Theorem cnvsn0 5251

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 4968	. . 3 |- dom { (/) } = ran `' { (/) }
2		dmsn0 5250	. . 3 |- dom { (/) } = (/)
3	1, 2	eqtr3i 2261	. 2 |- ran `' { (/) } = (/)
4		relcnv 5160	. . 3 |- Rel `' { (/) }
5		relrn0 5039	. . 3 |- ( Rel `' { (/) } -> ( `' { (/) } = (/) <-> ran `' { (/) } = (/) ) )
6	4, 5	ax-mp 5	. 2 |- ( `' { (/) } = (/) <-> ran `' { (/) } = (/) )
7	3, 6	mpbir 146	1 |- `' { (/) } = (/)

Syntax hints:   <-> wb 105   = wceq 1402   (/)c0 3520   {csn 3705   `'ccnv 4768   dom cdm 4769   ran crn 4770   Rel wrel 4774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  brtpos0  6513  tpostpos  6525