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Theorem sorpss 7434
Description: Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpss ( [] Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem sorpss
StepHypRef Expression
1 porpss 7433 . . 3 [] Po 𝐴
21biantrur 534 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ ( [] Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥)))
3 sspsstri 4030 . . . 4 ((𝑥𝑦𝑦𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
4 vex 3444 . . . . . 6 𝑦 ∈ V
54brrpss 7432 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
6 biid 264 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
7 vex 3444 . . . . . 6 𝑥 ∈ V
87brrpss 7432 . . . . 5 (𝑦 [] 𝑥𝑦𝑥)
95, 6, 83orbi123i 1153 . . . 4 ((𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
103, 9bitr4i 281 . . 3 ((𝑥𝑦𝑦𝑥) ↔ (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
11102ralbii 3134 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
12 df-so 5439 . 2 ( [] Or 𝐴 ↔ ( [] Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥)))
132, 11, 123bitr4ri 307 1 ( [] Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wo 844  w3o 1083  wral 3106  wss 3881  wpss 3882   class class class wbr 5030   Po wpo 5436   Or wor 5437   [] crpss 7428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-rpss 7429
This theorem is referenced by:  sorpsscmpl  7440  enfin2i  9732  fin1a2lem13  9823
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