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Theorem swopo 5457
 Description: A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
swopo.1 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
swopo.2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
Assertion
Ref Expression
swopo (𝜑𝑅 Po 𝐴)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem swopo
StepHypRef Expression
1 id 22 . . . . 5 (𝑥𝐴𝑥𝐴)
21ancli 552 . . . 4 (𝑥𝐴 → (𝑥𝐴𝑥𝐴))
3 swopo.1 . . . . 5 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
43ralrimivva 3120 . . . 4 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
5 breq1 5039 . . . . . 6 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
6 breq2 5040 . . . . . . 7 (𝑦 = 𝑥 → (𝑧𝑅𝑦𝑧𝑅𝑥))
76notbid 321 . . . . . 6 (𝑦 = 𝑥 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝑥))
85, 7imbi12d 348 . . . . 5 (𝑦 = 𝑥 → ((𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦) ↔ (𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥)))
9 breq2 5040 . . . . . 6 (𝑧 = 𝑥 → (𝑥𝑅𝑧𝑥𝑅𝑥))
10 breq1 5039 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑅𝑥𝑥𝑅𝑥))
1110notbid 321 . . . . . 6 (𝑧 = 𝑥 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑥𝑅𝑥))
129, 11imbi12d 348 . . . . 5 (𝑧 = 𝑥 → ((𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥) ↔ (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥)))
138, 12rspc2va 3554 . . . 4 (((𝑥𝐴𝑥𝐴) ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦)) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
142, 4, 13syl2anr 599 . . 3 ((𝜑𝑥𝐴) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
1514pm2.01d 193 . 2 ((𝜑𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1633adantr1 1166 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
17 swopo.2 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1817imp 410 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑥𝑅𝑧𝑧𝑅𝑦))
1918orcomd 868 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑧𝑅𝑦𝑥𝑅𝑧))
2019ord 861 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (¬ 𝑧𝑅𝑦𝑥𝑅𝑧))
2120expimpd 457 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 ∧ ¬ 𝑧𝑅𝑦) → 𝑥𝑅𝑧))
2216, 21sylan2d 607 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2315, 22ispod 5455 1 (𝜑𝑅 Po 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3070   class class class wbr 5036   Po wpo 5445 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 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-un 3865  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-po 5447 This theorem is referenced by:  swoer  8335  swoso  8338
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