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Theorem isoso 7369
Description: An isomorphism preserves the property of being a strict total order. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
isoso (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))

Proof of Theorem isoso
StepHypRef Expression
1 isocnv 7351 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2 isosolem 7368 . . 3 (𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → (𝑅 Or 𝐴𝑆 Or 𝐵))
31, 2syl 17 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))
4 isosolem 7368 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
53, 4impbid 212 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   Or wor 5590  ccnv 5683   Isom wiso 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569
This theorem is referenced by:  isowe  7370  supiso  9516  cnso  16284  wepwso  43060  rrx2plordso  48650
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