Theorem cnveq0 5122

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 5069	. 2 |- `' (/) = (/)
2		rel0 4784	. . . . 5 |- Rel (/)
3		cnveqb 5121	. . . . 5 |- ( ( Rel A /\ Rel (/) ) -> ( A = (/) <-> `' A = `' (/) ) )
4	2, 3	mpan2 425	. . . 4 |- ( Rel A -> ( A = (/) <-> `' A = `' (/) ) )
5		eqeq2 2203	. . . . 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 2196	. 2 |- ( `' (/) = (/) -> ( Rel A -> ( A = (/) <-> `' A = (/) ) ) )
9	1, 8	ax-mp 5	1 |- ( Rel A -> ( A = (/) <-> `' A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   (/)c0 3446   `'ccnv 4658   Rel wrel 4664

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667

This theorem is referenced by: (None)