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Theorem isso2i 5501
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 2012 . . . . 5 𝑥 = 𝑥
21orci 861 . . . 4 (𝑥 = 𝑥𝑥𝑅𝑥)
3 nfv 1908 . . . . 5 𝑦((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
4 eleq1w 2893 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
54anbi2d 630 . . . . . 6 (𝑦 = 𝑥 → ((𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴𝑥𝐴)))
6 equequ2 2026 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥 = 𝑦𝑥 = 𝑥))
7 breq1 5060 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝑅𝑥𝑥𝑅𝑥))
86, 7orbi12d 914 . . . . . . 7 (𝑦 = 𝑥 → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥 = 𝑥𝑥𝑅𝑥)))
9 breq2 5061 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
109notbid 320 . . . . . . 7 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
118, 10bibi12d 348 . . . . . 6 (𝑦 = 𝑥 → (((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦) ↔ ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥)))
125, 11imbi12d 347 . . . . 5 (𝑦 = 𝑥 → (((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦)) ↔ ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))))
13 isso2i.1 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦𝑦𝑅𝑥)))
1413con2bid 357 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦))
153, 12, 14chvarfv 2234 . . . 4 ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
162, 15mpbii 235 . . 3 ((𝑥𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1716anidms 569 . 2 (𝑥𝐴 → ¬ 𝑥𝑅𝑥)
18 isso2i.2 . 2 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1914biimprd 250 . . . 4 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 → (𝑥 = 𝑦𝑦𝑅𝑥)))
2019orrd 859 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
21 3orass 1084 . . 3 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
2220, 21sylibr 236 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2317, 18, 22issoi 5500 1 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3o 1080  w3a 1081  wcel 2107   class class class wbr 5057   Or wor 5466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-po 5467  df-so 5468
This theorem is referenced by:  ltsonq  10383  ltsosr  10508  ltso  10713  xrltso  12526
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