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Theorem isoso 6069
Description: An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
isoso  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
<->  S  Or  B ) )

Proof of Theorem isoso
StepHypRef Expression
1 isocnv 6051 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 isosolem 6068 . . 3  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( R  Or  A  ->  S  Or  B ) )
31, 2syl 16 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A  ->  S  Or  B
) )
4 isosolem 6068 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A
) )
53, 4impbid 185 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
<->  S  Or  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    Or wor 4503   `'ccnv 4878    Isom wiso 5456
This theorem is referenced by:  isowe  6070  supiso  7478  cnso  12847  wepwso  27118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464
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