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

Theorem isowe 5698
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 5691 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A 
<->  S  Fr  B ) )
2 isoso 5697 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
<->  S  Or  B ) )
31, 2anbi12d 694 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( ( R  Fr  A  /\  R  Or  A )  <->  ( S  Fr  B  /\  S  Or  B ) ) )
4 df-we 4247 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
5 df-we 4247 . 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 4206    Fr wfr 4242    We wwe 4244    Isom wiso 4593
This theorem is referenced by:  f1owe  5702  hartogslem1  7141  oemapwe  7280  om2uzoi  10896
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609
  Copyright terms: Public domain W3C validator