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Theorem dfwe2 6850
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 4989 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
2 df-so 4950 . . . 4 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
3 simpr 475 . . . . 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 5006 . . . . . . . . . . . . . . . . 17 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ¬ (𝑥𝑅𝑦𝑦𝑅𝑥))
763adantr3 1214 . . . . . . . . . . . . . . . 16 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ (𝑥𝑅𝑦𝑦𝑅𝑥))
8 breq2 4581 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
98anbi2d 735 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
109notbid 306 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (¬ (𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ¬ (𝑥𝑅𝑦𝑦𝑅𝑧)))
117, 10syl5ibcom 233 . . . . . . . . . . . . . . 15 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥 = 𝑧 → ¬ (𝑥𝑅𝑦𝑦𝑅𝑧)))
12 pm2.21 118 . . . . . . . . . . . . . . 15 (¬ (𝑥𝑅𝑦𝑦𝑅𝑧) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1311, 12syl6 34 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥 = 𝑧 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
14 fr3nr 6848 . . . . . . . . . . . . . . . . 17 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ (𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥))
15 df-3an 1032 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥))
1615biimpri 216 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑅𝑦𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥) → (𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥))
1716ancoms 467 . . . . . . . . . . . . . . . . 17 ((𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧𝑧𝑅𝑥))
1814, 17nsyl 133 . . . . . . . . . . . . . . . 16 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ (𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)))
1918pm2.21d 116 . . . . . . . . . . . . . . 15 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧))
2019expd 450 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑧𝑅𝑥 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
215, 13, 203jaod 1383 . . . . . . . . . . . . 13 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
22 frirr 5005 . . . . . . . . . . . . . 14 ((𝑅 Fr 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
23223ad2antr1 1218 . . . . . . . . . . . . 13 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ 𝑥𝑅𝑥)
2421, 23jctild 563 . . . . . . . . . . . 12 ((𝑅 Fr 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2524ex 448 . . . . . . . . . . 11 (𝑅 Fr 𝐴 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
2625a2d 29 . . . . . . . . . 10 (𝑅 Fr 𝐴 → (((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
2726alimdv 1831 . . . . . . . . 9 (𝑅 Fr 𝐴 → (∀𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)) → ∀𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
28272alimdv 1833 . . . . . . . 8 (𝑅 Fr 𝐴 → (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)) → ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
29 r3al 2923 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)))
30 r3al 2923 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
3128, 29, 303imtr4g 283 . . . . . . 7 (𝑅 Fr 𝐴 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
32 ralidm 4026 . . . . . . . . 9 (∀𝑦𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
33 breq2 4581 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑥𝑅𝑦𝑥𝑅𝑧))
34 equequ2 1939 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
35 breq1 4580 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
3633, 34, 353orbi123d 1389 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)))
3736cbvralv 3146 . . . . . . . . . 10 (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
3837ralbii 2962 . . . . . . . . 9 (∀𝑦𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
3932, 38bitr3i 264 . . . . . . . 8 (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
4039ralbii 2962 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
41 df-po 4949 . . . . . . 7 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
4231, 40, 413imtr4g 283 . . . . . 6 (𝑅 Fr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → 𝑅 Po 𝐴))
4342ancrd 574 . . . . 5 (𝑅 Fr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))))
443, 43impbid2 214 . . . 4 (𝑅 Fr 𝐴 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
452, 44syl5bb 270 . . 3 (𝑅 Fr 𝐴 → (𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
4645pm5.32i 666 . 2 ((𝑅 Fr 𝐴𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
471, 46bitri 262 1 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3o 1029  w3a 1030  wal 1472  wcel 1976  wral 2895   class class class wbr 4577   Po wpo 4947   Or wor 4948   Fr wfr 4984   We wwe 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-po 4949  df-so 4950  df-fr 4987  df-we 4989
This theorem is referenced by:  ordon  6851  f1oweALT  7020  dford2  8377  fpwwe2lem12  9319  fpwwe2lem13  9320  dfon2  30747  fnwe2  36437
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