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Theorem cnveq0 5389
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
StepHypRef Expression
1 cnv0 5335 . 2  |-  `' (/)  =  (/)
2 rel0 5059 . . . . 5  |-  Rel  (/)
3 cnveqb 5388 . . . . 5  |-  ( ( Rel  A  /\  Rel  (/) )  ->  ( A  =  (/)  <->  `' A  =  `' (/) ) )
42, 3mpan2 671 . . . 4  |-  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  `' (/) ) )
5 eqeq2 2465 . . . . 5  |-  ( (/)  =  `' (/)  ->  ( `' A  =  (/)  <->  `' A  =  `' (/) ) )
65bibi2d 318 . . . 4  |-  ( (/)  =  `' (/)  ->  ( ( A  =  (/)  <->  `' A  =  (/) )  <->  ( A  =  (/)  <->  `' A  =  `' (/) ) ) )
74, 6syl5ibr 221 . . 3  |-  ( (/)  =  `' (/)  ->  ( Rel  A  ->  ( A  =  (/) 
<->  `' A  =  (/) ) ) )
87eqcoms 2462 . 2  |-  ( `' (/)  =  (/)  ->  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  (/) ) ) )
91, 8ax-mp 5 1  |-  ( Rel 
A  ->  ( A  =  (/)  <->  `' A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   (/)c0 3732   `'ccnv 4934   Rel wrel 4940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-br 4388  df-opab 4446  df-xp 4941  df-rel 4942  df-cnv 4943
This theorem is referenced by:  elrn3  27704
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