Theorem cnvsn0 6161

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 5837	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6160	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2764	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6056	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5915	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 232	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 207   = wceq 1547  ∅c0 4261  {csn 4555  ^◡ccnv 5617  dom cdm 5618  ran crn 5619  Rel wrel 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629

This theorem is referenced by:  opswap  6180  brtpos0  8173  tpostpos  8186  dftpos5  49364