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Theorem swopo 5557
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 550 . . . 4 (𝑥𝐴 → (𝑥𝐴𝑥𝐴))
3 swopo.1 . . . . 5 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
43ralrimivva 3198 . . . 4 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
5 breq1 5109 . . . . . 6 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
6 breq2 5110 . . . . . . 7 (𝑦 = 𝑥 → (𝑧𝑅𝑦𝑧𝑅𝑥))
76notbid 318 . . . . . 6 (𝑦 = 𝑥 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝑥))
85, 7imbi12d 345 . . . . 5 (𝑦 = 𝑥 → ((𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦) ↔ (𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥)))
9 breq2 5110 . . . . . 6 (𝑧 = 𝑥 → (𝑥𝑅𝑧𝑥𝑅𝑥))
10 breq1 5109 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑅𝑥𝑥𝑅𝑥))
1110notbid 318 . . . . . 6 (𝑧 = 𝑥 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑥𝑅𝑥))
129, 11imbi12d 345 . . . . 5 (𝑧 = 𝑥 → ((𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥) ↔ (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥)))
138, 12rspc2va 3592 . . . 4 (((𝑥𝐴𝑥𝐴) ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦)) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
142, 4, 13syl2anr 598 . . 3 ((𝜑𝑥𝐴) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
1514pm2.01d 189 . 2 ((𝜑𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1633adantr1 1170 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
17 swopo.2 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1817imp 408 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑥𝑅𝑧𝑧𝑅𝑦))
1918orcomd 870 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑧𝑅𝑦𝑥𝑅𝑧))
2019ord 863 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (¬ 𝑧𝑅𝑦𝑥𝑅𝑧))
2120expimpd 455 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 ∧ ¬ 𝑧𝑅𝑦) → 𝑥𝑅𝑧))
2216, 21sylan2d 606 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2315, 22ispod 5555 1 (𝜑𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  w3a 1088  wcel 2107  wral 3065   class class class wbr 5106   Po wpo 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-po 5546
This theorem is referenced by:  swoer  8679  swoso  8682
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