Theorem cnvsn0 6209

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 5893	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6208	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2758	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6103	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5967	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 230	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 205   = wceq 1534  ∅c0 4319  {csn 4625  ^◡ccnv 5672  dom cdm 5673  ran crn 5674  Rel wrel 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-xp 5679  df-rel 5680  df-cnv 5681  df-dm 5683  df-rn 5684

This theorem is referenced by:  opswap  6228  brtpos0  8233  tpostpos  8246