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Theorem isso2i 5202
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 2096 . . . . 5 𝑥 = 𝑥
21orci 845 . . . 4 (𝑥 = 𝑥𝑥𝑅𝑥)
3 eleq1w 2832 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
43anbi2d 606 . . . . . 6 (𝑦 = 𝑥 → ((𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴𝑥𝐴)))
5 equequ2 2110 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥 = 𝑦𝑥 = 𝑥))
6 breq1 4787 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝑅𝑥𝑥𝑅𝑥))
75, 6orbi12d 883 . . . . . . 7 (𝑦 = 𝑥 → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥 = 𝑥𝑥𝑅𝑥)))
8 breq2 4788 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
98notbid 307 . . . . . . 7 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
107, 9bibi12d 334 . . . . . 6 (𝑦 = 𝑥 → (((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦) ↔ ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥)))
114, 10imbi12d 333 . . . . 5 (𝑦 = 𝑥 → (((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦)) ↔ ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))))
12 isso2i.1 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦𝑦𝑅𝑥)))
1312con2bid 343 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦))
1411, 13chvarv 2424 . . . 4 ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
152, 14mpbii 223 . . 3 ((𝑥𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1615anidms 548 . 2 (𝑥𝐴 → ¬ 𝑥𝑅𝑥)
17 isso2i.2 . 2 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1813biimprd 238 . . . 4 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 → (𝑥 = 𝑦𝑦𝑅𝑥)))
1918orrd 843 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
20 3orass 1073 . . 3 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
2119, 20sylibr 224 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2216, 17, 21issoi 5201 1 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826  w3o 1069  w3a 1070  wcel 2144   class class class wbr 4784   Or wor 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-po 5170  df-so 5171
This theorem is referenced by:  ltsonq  9992  ltsosr  10116  ltso  10319  xrltso  12178
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