MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  swopo Structured version   Visualization version   GIF version

Theorem swopo 5181
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 538 . . . 4 (𝑥𝐴 → (𝑥𝐴𝑥𝐴))
3 swopo.1 . . . . 5 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
43ralrimivva 3120 . . . 4 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
5 breq1 4790 . . . . . 6 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
6 breq2 4791 . . . . . . 7 (𝑦 = 𝑥 → (𝑧𝑅𝑦𝑧𝑅𝑥))
76notbid 307 . . . . . 6 (𝑦 = 𝑥 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝑥))
85, 7imbi12d 333 . . . . 5 (𝑦 = 𝑥 → ((𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦) ↔ (𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥)))
9 breq2 4791 . . . . . 6 (𝑧 = 𝑥 → (𝑥𝑅𝑧𝑥𝑅𝑥))
10 breq1 4790 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝑅𝑥𝑥𝑅𝑥))
1110notbid 307 . . . . . 6 (𝑧 = 𝑥 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑥𝑅𝑥))
129, 11imbi12d 333 . . . . 5 (𝑧 = 𝑥 → ((𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥) ↔ (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥)))
138, 12rspc2va 3473 . . . 4 (((𝑥𝐴𝑥𝐴) ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦)) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
142, 4, 13syl2anr 584 . . 3 ((𝜑𝑥𝐴) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
1514pm2.01d 181 . 2 ((𝜑𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1633adantr1 1174 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
17 swopo.2 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1817imp 393 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑥𝑅𝑧𝑧𝑅𝑦))
1918orcomd 860 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (𝑧𝑅𝑦𝑥𝑅𝑧))
2019ord 853 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑦) → (¬ 𝑧𝑅𝑦𝑥𝑅𝑧))
2120expimpd 441 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 ∧ ¬ 𝑧𝑅𝑦) → 𝑥𝑅𝑧))
2216, 21sylan2d 592 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2315, 22ispod 5179 1 (𝜑𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 836  w3a 1071  wcel 2145  wral 3061   class class class wbr 4787   Po wpo 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-po 5171
This theorem is referenced by:  swoer  7930  swoso  7933
  Copyright terms: Public domain W3C validator