Theorem cnvsn0 6169

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 5845	. . 3 ⊢ dom {∅} = ran ◡{∅}
2		dmsn0 6168	. . 3 ⊢ dom {∅} = ∅
3	1, 2	eqtr3i 2762	. 2 ⊢ ran ◡{∅} = ∅
4		relcnv 6064	. . 3 ⊢ Rel ◡{∅}
5		relrn0 5923	. . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅))
6	4, 5	ax-mp 5	. 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)
7	3, 6	mpbir 231	1 ⊢ ◡{∅} = ∅

Syntax hints:   ↔ wb 206   = wceq 1542  ∅c0 4274  {csn 4568  ^◡ccnv 5624  dom cdm 5625  ran crn 5626  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636

This theorem is referenced by:  opswap  6188  brtpos0  8177  tpostpos  8190  dftpos5  49364