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Theorem swopo 5241
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 545 . . . 4 (𝑥𝐴 → (𝑥𝐴𝑥𝐴))
3 swopo.1 . . . . 5 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
43ralrimivva 3150 . . . 4 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
5 breq1 4844 . . . . . 6 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
6 breq2 4845 . . . . . . 7 (𝑦 = 𝑥 → (𝑧𝑅𝑦𝑧𝑅𝑥))
76notbid 310 . . . . . 6 (𝑦 = 𝑥 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝑥))
85, 7imbi12d 336 . . . . 5 (𝑦 = 𝑥 → ((𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦) ↔ (𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥)))
9 breq2 4845 . . . . . 6 (𝑧 = 𝑥 → (𝑥𝑅𝑧𝑥𝑅𝑥))
10 breq1 4844 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑅𝑥𝑥𝑅𝑥))
1110notbid 310 . . . . . 6 (𝑧 = 𝑥 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑥𝑅𝑥))
129, 11imbi12d 336 . . . . 5 (𝑧 = 𝑥 → ((𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥) ↔ (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥)))
138, 12rspc2va 3509 . . . 4 (((𝑥𝐴𝑥𝐴) ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦)) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
142, 4, 13syl2anr 591 . . 3 ((𝜑𝑥𝐴) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
1514pm2.01d 182 . 2 ((𝜑𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1633adantr1 1211 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
17 swopo.2 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1817imp 396 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑥𝑅𝑧𝑧𝑅𝑦))
1918orcomd 898 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑧𝑅𝑦𝑥𝑅𝑧))
2019ord 891 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (¬ 𝑧𝑅𝑦𝑥𝑅𝑧))
2120expimpd 446 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 ∧ ¬ 𝑧𝑅𝑦) → 𝑥𝑅𝑧))
2216, 21sylan2d 599 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2315, 22ispod 5239 1 (𝜑𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874  w3a 1108  wcel 2157  wral 3087   class class class wbr 4841   Po wpo 5229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-po 5231
This theorem is referenced by:  swoer  8010  swoso  8013
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