Theorem cnvsn0 6211

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 5877	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6210	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2786	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6100	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5955	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 234	1 ⊢ ◡{∅} = ∅

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∅c0 4279  {csn 4584  ^◡ccnv 5650  dom cdm 5651  ran crn 5652  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662

This theorem is used by:  opswap  6230  brtpos0  8250  tpostpos  8263  dftpos5  49981