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Theorem dfwe2 7480
 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 5484 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
2 df-so 5443 . . . 4 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
3 simpr 488 . . . . 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 5501 . . . . . . . . . . . . . . . . 17 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ¬ (𝑥𝑅𝑦𝑦𝑅𝑥))
763adantr3 1168 . . . . . . . . . . . . . . . 16 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ (𝑥𝑅𝑦𝑦𝑅𝑥))
8 breq2 5037 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
98anbi2d 631 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
109notbid 321 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (¬ (𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ¬ (𝑥𝑅𝑦𝑦𝑅𝑧)))
117, 10syl5ibcom 248 . . . . . . . . . . . . . . 15 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥 = 𝑧 → ¬ (𝑥𝑅𝑦𝑦𝑅𝑧)))
12 pm2.21 123 . . . . . . . . . . . . . . 15 (¬ (𝑥𝑅𝑦𝑦𝑅𝑧) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1311, 12syl6 35 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥 = 𝑧 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
14 fr3nr 7478 . . . . . . . . . . . . . . . . 17 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ (𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥))
15 df-3an 1086 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥))
1615biimpri 231 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑅𝑦𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥) → (𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥))
1716ancoms 462 . . . . . . . . . . . . . . . . 17 ((𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥))
1814, 17nsyl 142 . . . . . . . . . . . . . . . 16 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ (𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)))
1918pm2.21d 121 . . . . . . . . . . . . . . 15 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧))
2019expd 419 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑧𝑅𝑥 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
215, 13, 203jaod 1425 . . . . . . . . . . . . 13 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
22 frirr 5500 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
23223ad2antr1 1185 . . . . . . . . . . . . 13 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ 𝑥𝑅𝑥)
2421, 23jctild 529 . . . . . . . . . . . 12 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2524ex 416 . . . . . . . . . . 11 (𝑅 Fr 𝐴 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
2625a2d 29 . . . . . . . . . 10 (𝑅 Fr 𝐴 → (((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
2726alimdv 1917 . . . . . . . . 9 (𝑅 Fr 𝐴 → (∀𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)) → ∀𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
28272alimdv 1919 . . . . . . . 8 (𝑅 Fr 𝐴 → (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)) → ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
29 r3al 3170 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)))
30 r3al 3170 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
3128, 29, 303imtr4g 299 . . . . . . 7 (𝑅 Fr 𝐴 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
32 ralidm 4416 . . . . . . . . 9 (∀𝑦𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
33 breq2 5037 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑥𝑅𝑦𝑥𝑅𝑧))
34 equequ2 2033 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
35 breq1 5036 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
3633, 34, 353orbi123d 1432 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)))
3736cbvralvw 3399 . . . . . . . . . 10 (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
3837ralbii 3136 . . . . . . . . 9 (∀𝑦𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
3932, 38bitr3i 280 . . . . . . . 8 (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
4039ralbii 3136 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
41 df-po 5442 . . . . . . 7 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
4231, 40, 413imtr4g 299 . . . . . 6 (𝑅 Fr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → 𝑅 Po 𝐴))
4342ancrd 555 . . . . 5 (𝑅 Fr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))))
443, 43impbid2 229 . . . 4 (𝑅 Fr 𝐴 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
452, 44syl5bb 286 . . 3 (𝑅 Fr 𝐴 → (𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
4645pm5.32i 578 . 2 ((𝑅 Fr 𝐴𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
471, 46bitri 278 1 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084  ∀wal 1536   ∈ wcel 2112  ∀wral 3109   class class class wbr 5033   Po wpo 5440   Or wor 5441   Fr wfr 5479   We wwe 5481 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-po 5442  df-so 5443  df-fr 5482  df-we 5484 This theorem is referenced by:  epweon  7481  f1oweALT  7659  dford2  9071  fpwwe2lem12  10056  fpwwe2lem13  10057  dfon2  33145  fnwe2  39984
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