Theorem cnveq0 4797

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 4747	. 2 |- `' (/) = (/)
2		rel0 4480	. . . . 5 |- Rel (/)
3		cnveqb 4796	. . . . 5 |- ( ( Rel A /\ Rel (/) ) -> ( A = (/) <-> `' A = `' (/) ) )
4	2, 3	mpan2 415	. . . 4 |- ( Rel A -> ( A = (/) <-> `' A = `' (/) ) )
5		eqeq2 2090	. . . . 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 2084	. 2 |- ( `' (/) = (/) -> ( Rel A -> ( A = (/) <-> `' A = (/) ) ) )
9	1, 8	ax-mp 7	1 |- ( Rel A -> ( A = (/) <-> `' A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   (/)c0 3251   `'ccnv 4362   Rel wrel 4368

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371

This theorem is referenced by: (None)