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Theorem dfwe2 7023
Description: Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
dfwe2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝐴,𝑦

Proof of Theorem dfwe2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-we 5104 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
2 df-so 5065 . . . 4 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
3 simpr 476 . . . . 5 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
4 ax-1 6 . . . . . . . . . . . . . . 15 (𝑥𝑅𝑧 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
54a1i 11 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑧 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
6 fr2nr 5121 . . . . . . . . . . . . . . . . 17 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ¬ (𝑥𝑅𝑦𝑦𝑅𝑥))
763adantr3 1242 . . . . . . . . . . . . . . . 16 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ (𝑥𝑅𝑦𝑦𝑅𝑥))
8 breq2 4689 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
98anbi2d 740 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
109notbid 307 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (¬ (𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ¬ (𝑥𝑅𝑦𝑦𝑅𝑧)))
117, 10syl5ibcom 235 . . . . . . . . . . . . . . 15 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥 = 𝑧 → ¬ (𝑥𝑅𝑦𝑦𝑅𝑧)))
12 pm2.21 120 . . . . . . . . . . . . . . 15 (¬ (𝑥𝑅𝑦𝑦𝑅𝑧) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1311, 12syl6 35 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥 = 𝑧 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
14 fr3nr 7021 . . . . . . . . . . . . . . . . 17 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ (𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥))
15 df-3an 1056 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥))
1615biimpri 218 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑅𝑦𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥) → (𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥))
1716ancoms 468 . . . . . . . . . . . . . . . . 17 ((𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥))
1814, 17nsyl 135 . . . . . . . . . . . . . . . 16 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ (𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)))
1918pm2.21d 118 . . . . . . . . . . . . . . 15 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧))
2019expd 451 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑧𝑅𝑥 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
215, 13, 203jaod 1432 . . . . . . . . . . . . 13 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
22 frirr 5120 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
23223ad2antr1 1246 . . . . . . . . . . . . 13 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ 𝑥𝑅𝑥)
2421, 23jctild 565 . . . . . . . . . . . 12 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2524ex 449 . . . . . . . . . . 11 (𝑅 Fr 𝐴 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
2625a2d 29 . . . . . . . . . 10 (𝑅 Fr 𝐴 → (((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
2726alimdv 1885 . . . . . . . . 9 (𝑅 Fr 𝐴 → (∀𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)) → ∀𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
28272alimdv 1887 . . . . . . . 8 (𝑅 Fr 𝐴 → (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)) → ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
29 r3al 2969 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)))
30 r3al 2969 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
3128, 29, 303imtr4g 285 . . . . . . 7 (𝑅 Fr 𝐴 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
32 ralidm 4108 . . . . . . . . 9 (∀𝑦𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
33 breq2 4689 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑥𝑅𝑦𝑥𝑅𝑧))
34 equequ2 1999 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
35 breq1 4688 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
3633, 34, 353orbi123d 1438 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)))
3736cbvralv 3201 . . . . . . . . . 10 (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
3837ralbii 3009 . . . . . . . . 9 (∀𝑦𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
3932, 38bitr3i 266 . . . . . . . 8 (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
4039ralbii 3009 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
41 df-po 5064 . . . . . . 7 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
4231, 40, 413imtr4g 285 . . . . . 6 (𝑅 Fr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → 𝑅 Po 𝐴))
4342ancrd 576 . . . . 5 (𝑅 Fr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))))
443, 43impbid2 216 . . . 4 (𝑅 Fr 𝐴 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
452, 44syl5bb 272 . . 3 (𝑅 Fr 𝐴 → (𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
4645pm5.32i 670 . 2 ((𝑅 Fr 𝐴𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
471, 46bitri 264 1 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3o 1053  w3a 1054  wal 1521  wcel 2030  wral 2941   class class class wbr 4685   Po wpo 5062   Or wor 5063   Fr wfr 5099   We wwe 5101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-po 5064  df-so 5065  df-fr 5102  df-we 5104
This theorem is referenced by:  ordon  7024  f1oweALT  7194  dford2  8555  fpwwe2lem12  9501  fpwwe2lem13  9502  dfon2  31821  fnwe2  37940
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