Theorem cnveq0 5158

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 5105	. 2 |- `' (/) = (/)
2		rel0 4818	. . . . 5 |- Rel (/)
3		cnveqb 5157	. . . . 5 |- ( ( Rel A /\ Rel (/) ) -> ( A = (/) <-> `' A = `' (/) ) )
4	2, 3	mpan2 425	. . . 4 |- ( Rel A -> ( A = (/) <-> `' A = `' (/) ) )
5		eqeq2 2217	. . . . 5 |- ( (/) = `' (/) -> ( `' A = (/) <-> `' A = `' (/) ) )
6	5	bibi2d 232	. . . 4 |- ( (/) = `' (/) -> ( ( A = (/) <-> `' A = (/) ) <-> ( A = (/) <-> `' A = `' (/) ) ) )
7	4, 6	imbitrrid 156	. . 3 |- ( (/) = `' (/) -> ( Rel A -> ( A = (/) <-> `' A = (/) ) ) )
8	7	eqcoms 2210	. 2 |- ( `' (/) = (/) -> ( Rel A -> ( A = (/) <-> `' A = (/) ) ) )
9	1, 8	ax-mp 5	1 |- ( Rel A -> ( A = (/) <-> `' A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   (/)c0 3468   `'ccnv 4692   Rel wrel 4698

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701

This theorem is referenced by: (None)