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Theorem isso2i 5538
Description: Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
isso2i.1 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦𝑦𝑅𝑥)))
isso2i.2 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Assertion
Ref Expression
isso2i 𝑅 Or 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧

Proof of Theorem isso2i
StepHypRef Expression
1 equid 2015 . . . . 5 𝑥 = 𝑥
21orci 862 . . . 4 (𝑥 = 𝑥𝑥𝑅𝑥)
3 nfv 1917 . . . . 5 𝑦((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
4 eleq1w 2821 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
54anbi2d 629 . . . . . 6 (𝑦 = 𝑥 → ((𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴𝑥𝐴)))
6 equequ2 2029 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥 = 𝑦𝑥 = 𝑥))
7 breq1 5077 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝑅𝑥𝑥𝑅𝑥))
86, 7orbi12d 916 . . . . . . 7 (𝑦 = 𝑥 → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥 = 𝑥𝑥𝑅𝑥)))
9 breq2 5078 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
109notbid 318 . . . . . . 7 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
118, 10bibi12d 346 . . . . . 6 (𝑦 = 𝑥 → (((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦) ↔ ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥)))
125, 11imbi12d 345 . . . . 5 (𝑦 = 𝑥 → (((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦)) ↔ ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))))
13 isso2i.1 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦𝑦𝑅𝑥)))
1413con2bid 355 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦))
153, 12, 14chvarfv 2233 . . . 4 ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
162, 15mpbii 232 . . 3 ((𝑥𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1716anidms 567 . 2 (𝑥𝐴 → ¬ 𝑥𝑅𝑥)
18 isso2i.2 . 2 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1914biimprd 247 . . . 4 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 → (𝑥 = 𝑦𝑦𝑅𝑥)))
2019orrd 860 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
21 3orass 1089 . . 3 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
2220, 21sylibr 233 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2317, 18, 22issoi 5537 1 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844  w3o 1085  w3a 1086  wcel 2106   class class class wbr 5074   Or wor 5502
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-po 5503  df-so 5504
This theorem is referenced by:  ltsonq  10725  ltsosr  10850  ltso  11055  xrltso  12875
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