Theorem cnvsn0 6198

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 5872	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6197	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2788	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6094	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5950	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 233	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 208   = wceq 1561  ∅c0 4286  {csn 4583  ^◡ccnv 5647  dom cdm 5648  ran crn 5649  Rel wrel 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-rel 5655  df-cnv 5656  df-dm 5658  df-rn 5659

This theorem is referenced by:  opswap  6217  brtpos0  8214  tpostpos  8227  dftpos5  49496