Theorem cnvsn0 5820

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 5520	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 5819	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2824	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 5721	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5588	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 223	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 198   = wceq 1653  ∅c0 4116  {csn 4369  ^◡ccnv 5312  dom cdm 5313  ran crn 5314  Rel wrel 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-br 4845  df-opab 4907  df-xp 5319  df-rel 5320  df-cnv 5321  df-dm 5323  df-rn 5324

This theorem is referenced by:  opswap  5842  brtpos0  7598  tpostpos  7611