Theorem cnvsn0 6166

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 5842	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6165	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2759	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6061	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5920	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 231	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 206   = wceq 1541  ∅c0 4283  {csn 4578  ^◡ccnv 5621  dom cdm 5622  ran crn 5623  Rel wrel 5627

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 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633

This theorem is referenced by:  opswap  6185  brtpos0  8173  tpostpos  8186  dftpos5  49061