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Theorem isowe 6045
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 6038 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A 
<->  S  Fr  B ) )
2 isoso 6044 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
<->  S  Or  B ) )
31, 2anbi12d 710 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( ( R  Fr  A  /\  R  Or  A )  <->  ( S  Fr  B  /\  S  Or  B ) ) )
4 df-we 4686 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
5 df-we 4686 . 2  |-  ( S  We  B  <->  ( S  Fr  B  /\  S  Or  B ) )
63, 4, 53bitr4g 288 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  We  A 
<->  S  We  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    Or wor 4645    Fr wfr 4681    We wwe 4683    Isom wiso 5424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432
This theorem is referenced by:  f1owe  6049  hartogslem1  7761  oemapwe  7907  oemapweOLD  7929  om2uzoi  11783
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