Theorem cnvsn0 5749

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 5459	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 5748	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2772	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 5649	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5526	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 221	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 196   = wceq 1620  ∅c0 4046  {csn 4309  ^◡ccnv 5253  dom cdm 5254  ran crn 5255  Rel wrel 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-rel 5261  df-cnv 5262  df-dm 5264  df-rn 5265

This theorem is referenced by:  opswap  5771  brtpos0  7516  tpostpos  7529