MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isoso Structured version   Unicode version

Theorem isoso 6225
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 6207 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 isosolem 6224 . . 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 6224 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A
) )
53, 4impbid 191 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
<->  S  Or  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    Or wor 4794   `'ccnv 4993    Isom wiso 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590
This theorem is referenced by:  isowe  6226  supiso  7924  cnso  13832  wepwso  30583
  Copyright terms: Public domain W3C validator