Theorem 0trrel 15054

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 6258	. 2 ⊢ (∅ ∘ ∅) = ∅
2	1	eqimssi 3991	1 ⊢ (∅ ∘ ∅) ⊆ ∅

This proof depends on syntax axioms:   ⊆ wss 3899  ∅c0 4279   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664

This theorem is used by:  trclfvcotrg  15089  ust0  24446