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Theorem swopo 4070
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 19 . . . . 5 (𝑥𝐴𝑥𝐴)
21ancli 310 . . . 4 (𝑥𝐴 → (𝑥𝐴𝑥𝐴))
3 swopo.1 . . . . 5 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
43ralrimivva 2418 . . . 4 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
5 breq1 3794 . . . . . 6 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
6 breq2 3795 . . . . . . 7 (𝑦 = 𝑥 → (𝑧𝑅𝑦𝑧𝑅𝑥))
76notbid 602 . . . . . 6 (𝑦 = 𝑥 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝑥))
85, 7imbi12d 227 . . . . 5 (𝑦 = 𝑥 → ((𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦) ↔ (𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥)))
9 breq2 3795 . . . . . 6 (𝑧 = 𝑥 → (𝑥𝑅𝑧𝑥𝑅𝑥))
10 breq1 3794 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑅𝑥𝑥𝑅𝑥))
1110notbid 602 . . . . . 6 (𝑧 = 𝑥 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑥𝑅𝑥))
129, 11imbi12d 227 . . . . 5 (𝑧 = 𝑥 → ((𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥) ↔ (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥)))
138, 12rspc2va 2685 . . . 4 (((𝑥𝐴𝑥𝐴) ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦)) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
142, 4, 13syl2anr 278 . . 3 ((𝜑𝑥𝐴) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
1514pm2.01d 558 . 2 ((𝜑𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1633adantr1 1074 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
17 swopo.2 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1817imp 119 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑥𝑅𝑧𝑧𝑅𝑦))
1918orcomd 658 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑧𝑅𝑦𝑥𝑅𝑧))
2019ord 653 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (¬ 𝑧𝑅𝑦𝑥𝑅𝑧))
2120expimpd 349 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 ∧ ¬ 𝑧𝑅𝑦) → 𝑥𝑅𝑧))
2216, 21sylan2d 282 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2315, 22ispod 4068 1 (𝜑𝑅 Po 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  w3a 896  wcel 1409  wral 2323   class class class wbr 3791   Po wpo 4058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-po 4060
This theorem is referenced by:  swoer  6164
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