Theorem cnvsn0 6183

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 5859	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6182	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2754	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6075	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5936	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 231	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 206   = wceq 1540  ∅c0 4296  {csn 4589  ^◡ccnv 5637  dom cdm 5638  ran crn 5639  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649

This theorem is referenced by:  opswap  6202  brtpos0  8212  tpostpos  8225  dftpos5  48862