Theorem 0trrel 14947

Description: The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.)

Assertion

Ref	Expression
0trrel	⊢ (∅ ∘ ∅) ⊆ ∅

Proof of Theorem 0trrel

Step	Hyp	Ref	Expression
1		co01 6234	. 2 ⊢ (∅ ∘ ∅) = ∅
2	1	eqimssi 4007	1 ⊢ (∅ ∘ ∅) ⊆ ∅

Syntax hints:   ⊆ wss 3914  ∅c0 4296   ∘ ccom 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647

This theorem is referenced by:  trclfvcotrg  14982  ust0  24107