Theorem cnveq0 4828

Description: A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
cnveq0	|- ( Rel A -> ( A = (/) <-> `' A = (/) ) )

Proof of Theorem cnveq0

Step	Hyp	Ref	Expression
1		cnv0 4778	. 2 |- `' (/) = (/)
2		rel0 4511	. . . . 5 |- Rel (/)
3		cnveqb 4827	. . . . 5 |- ( ( Rel A /\ Rel (/) ) -> ( A = (/) <-> `' A = `' (/) ) )
4	2, 3	mpan2 416	. . . 4 |- ( Rel A -> ( A = (/) <-> `' A = `' (/) ) )
5		eqeq2 2092	. . . . 5 |- ( (/) = `' (/) -> ( `' A = (/) <-> `' A = `' (/) ) )
6	5	bibi2d 230	. . . 4 |- ( (/) = `' (/) -> ( ( A = (/) <-> `' A = (/) ) <-> ( A = (/) <-> `' A = `' (/) ) ) )
7	4, 6	syl5ibr 154	. . 3 |- ( (/) = `' (/) -> ( Rel A -> ( A = (/) <-> `' A = (/) ) ) )
8	7	eqcoms 2086	. 2 |- ( `' (/) = (/) -> ( Rel A -> ( A = (/) <-> `' A = (/) ) ) )
9	1, 8	ax-mp 7	1 |- ( Rel A -> ( A = (/) <-> `' A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   (/)c0 3268   `'ccnv 4391   Rel wrel 4397

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-rel 4399  df-cnv 4400

This theorem is referenced by: (None)