Try just using the traditional $i
iterator (untested) but I can’t see why this wouldn’t work..
<?php
$i = 0;
while ( have_posts() ) : the_post();
$i++;
if ( $i == 1 ): ?>
<!-- First Post -->
<div id="post">
<a class="post_image" href="https://wordpress.stackexchange.com/questions/165568/<?php the_permalink(); ?>" title="<?php the_title(); ?>">
<?php if ( has_post_thumbnail() ): ?>
<?php the_post_thumbnail('full'); ?>
<?php else : ?>
<img src="<?php echo get_template_directory_uri(); ?>/images/No-Thumbnail-Available.png" alt="<?php the_title(); ?>" />
<?php endif; ?>
</a>
<h2 class="post_title"><a href="https://wordpress.stackexchange.com/questions/165568/<?php the_permalink(); ?>" rel="bookmark" title="<?php the_title(); ?>"><?php the_title(); ?></a></h2>
<span class="post_desc"><?php the_excerpt(); ?></span>
<div class="clear"></div>
</div>
<?php else: // else $i == 1 ?>
<!-- All Other Posts -->
<div id="post">
<a class="post_image" href="https://wordpress.stackexchange.com/questions/165568/<?php the_permalink(); ?>" title="<?php the_title(); ?>">
<?php if ( has_post_thumbnail() ): ?>
<?php the_post_thumbnail('thumbnail'); ?>
<?php else : ?>
<img src="<?php echo get_template_directory_uri(); ?>/images/No-Thumbnail-Available.png" alt="<?php the_title(); ?>" />
<?php endif; ?>
</a>
<h2 class="post_title"><a href="https://wordpress.stackexchange.com/questions/165568/<?php the_permalink(); ?>" rel="bookmark" title="<?php the_title(); ?>"><?php the_title(); ?></a></h2>
<span class="post_desc"><?php the_excerpt(); ?></span>
<div class="clear"></div>
</div>
<?php endif; // endif for $i == 1 ?>
<?php endwhile; ?>