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Theorem isso2i 5230
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 2109 . . . . 5 𝑥 = 𝑥
21orci 891 . . . 4 (𝑥 = 𝑥𝑥𝑅𝑥)
3 eleq1w 2827 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
43anbi2d 622 . . . . . 6 (𝑦 = 𝑥 → ((𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴𝑥𝐴)))
5 equequ2 2123 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥 = 𝑦𝑥 = 𝑥))
6 breq1 4812 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝑅𝑥𝑥𝑅𝑥))
75, 6orbi12d 942 . . . . . . 7 (𝑦 = 𝑥 → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥 = 𝑥𝑥𝑅𝑥)))
8 breq2 4813 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
98notbid 309 . . . . . . 7 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
107, 9bibi12d 336 . . . . . 6 (𝑦 = 𝑥 → (((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦) ↔ ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥)))
114, 10imbi12d 335 . . . . 5 (𝑦 = 𝑥 → (((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦)) ↔ ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))))
12 isso2i.1 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦𝑦𝑅𝑥)))
1312con2bid 345 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦))
1411, 13chvarv 2369 . . . 4 ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
152, 14mpbii 224 . . 3 ((𝑥𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1615anidms 562 . 2 (𝑥𝐴 → ¬ 𝑥𝑅𝑥)
17 isso2i.2 . 2 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1813biimprd 239 . . . 4 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 → (𝑥 = 𝑦𝑦𝑅𝑥)))
1918orrd 889 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
20 3orass 1110 . . 3 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
2119, 20sylibr 225 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2216, 17, 21issoi 5229 1 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3o 1106  w3a 1107  wcel 2155   class class class wbr 4809   Or wor 5197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-po 5198  df-so 5199
This theorem is referenced by:  ltsonq  10044  ltsosr  10168  ltso  10372  xrltso  12174
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