Theorem cnvsn0 6165

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 5844	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6164	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2766	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6063	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5922	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 233	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 208   = wceq 1548  ∅c0 4264  {csn 4558  ^◡ccnv 5620  dom cdm 5621  ran crn 5622  Rel wrel 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632

This theorem is referenced by:  opswap  6184  brtpos0  8177  tpostpos  8190  dftpos5  49378