Theorem cnvsn0 6159

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 5838	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6158	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2754	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6055	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5914	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 231	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 206   = wceq 1540  ∅c0 4284  {csn 4577  ^◡ccnv 5618  dom cdm 5619  ran crn 5620  Rel wrel 5624

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 5235  ax-nul 5245  ax-pr 5371

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 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630

This theorem is referenced by:  opswap  6178  brtpos0  8166  tpostpos  8179  dftpos5  48862