Theorem cnveq0 5239

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 5186	. 2 |- `' (/) = (/)
2		rel0 4897	. . . . 5 |- Rel (/)
3		cnveqb 5238	. . . . 5 |- ( ( Rel A /\ Rel (/) ) -> ( A = (/) <-> `' A = `' (/) ) )
4	2, 3	mpan2 429	. . . 4 |- ( Rel A -> ( A = (/) <-> `' A = `' (/) ) )
5		eqeq2 2248	. . . . 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 2241	. 2 |- ( `' (/) = (/) -> ( Rel A -> ( A = (/) <-> `' A = (/) ) ) )
9	1, 8	ax-mp 5	1 |- ( Rel A -> ( A = (/) <-> `' A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   (/)c0 3520   `'ccnv 4768   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-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  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

This theorem is referenced by: (None)