Theorem cnvsn0 5205

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 4923	. . 3 |- dom { (/) } = ran `' { (/) }
2		dmsn0 5204	. . 3 |- dom { (/) } = (/)
3	1, 2	eqtr3i 2254	. 2 |- ran `' { (/) } = (/)
4		relcnv 5114	. . 3 |- Rel `' { (/) }
5		relrn0 4994	. . 3 |- ( Rel `' { (/) } -> ( `' { (/) } = (/) <-> ran `' { (/) } = (/) ) )
6	4, 5	ax-mp 5	. 2 |- ( `' { (/) } = (/) <-> ran `' { (/) } = (/) )
7	3, 6	mpbir 146	1 |- `' { (/) } = (/)

Syntax hints:   <-> wb 105   = wceq 1397   (/)c0 3494   {csn 3669   `'ccnv 4724   dom cdm 4725   ran crn 4726   Rel wrel 4730

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  brtpos0  6417  tpostpos  6429