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Theorem isowe 5808
Description: An isomorphism preserves well ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
isowe  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  We  A 
<->  S  We  B ) )

Proof of Theorem isowe
StepHypRef Expression
1 isofr 5801 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A 
<->  S  Fr  B ) )
2 isoso 5807 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
<->  S  Or  B ) )
31, 2anbi12d 693 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( ( R  Fr  A  /\  R  Or  A )  <->  ( S  Fr  B  /\  S  Or  B ) ) )
4 df-we 4354 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
5 df-we 4354 . 2  |-  ( S  We  B  <->  ( S  Fr  B  /\  S  Or  B ) )
63, 4, 53bitr4g 281 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  We  A 
<->  S  We  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    Or wor 4313    Fr wfr 4349    We wwe 4351    Isom wiso 5223
This theorem is referenced by:  f1owe  5812  hartogslem1  7253  oemapwe  7392  om2uzoi  11013
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231
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