Theorem 0trrel 15143

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 6266	. 2 ⊢ (∅ ∘ ∅) = ∅
2	1	eqimssi 3969	1 ⊢ (∅ ∘ ∅) ⊆ ∅

This proof depends on syntax axioms:   ⊆ wss 3896  ∅c0 4277   ∘ ccom 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660

This theorem is used by:  trclfvcotrg  15178  ust0  24548