Theorem cnvsn0 6218

Description: The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
cnvsn0	⊢ ◡{∅} = ∅

Proof of Theorem cnvsn0

Step	Hyp	Ref	Expression
1		dfdm4 5882	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6217	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2788	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6106	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5960	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 234	1 ⊢ ◡{∅} = ∅

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∅c0 4281  {csn 4588  ^◡ccnv 5654  dom cdm 5655  ran crn 5656  Rel wrel 5660

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  opswap  6237  brtpos0  8264  tpostpos  8277  dftpos5  50129