Theorem 0trrel 14883

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 6204	. 2 ⊢ (∅ ∘ ∅) = ∅
2	1	eqimssi 3990	1 ⊢ (∅ ∘ ∅) ⊆ ∅

Syntax hints:   ⊆ wss 3897  ∅c0 4278   ∘ ccom 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620

This theorem is referenced by:  trclfvcotrg  14918  ust0  24130