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Theorem sorpss 6840
 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 6839 . . 3 [] Po 𝐴
21biantrur 526 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ ( [] Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥)))
3 sspsstri 3671 . . . 4 ((𝑥𝑦𝑦𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
4 vex 3176 . . . . . 6 𝑦 ∈ V
54brrpss 6838 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
6 biid 250 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
7 vex 3176 . . . . . 6 𝑥 ∈ V
87brrpss 6838 . . . . 5 (𝑦 [] 𝑥𝑦𝑥)
95, 6, 83orbi123i 1245 . . . 4 ((𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
103, 9bitr4i 266 . . 3 ((𝑥𝑦𝑦𝑥) ↔ (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
11102ralbii 2964 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
12 df-so 4960 . 2 ( [] Or 𝐴 ↔ ( [] Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥)))
132, 11, 123bitr4ri 292 1 ( [] Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030  ∀wral 2896   ⊆ wss 3540   ⊊ wpss 3541   class class class wbr 4583   Po wpo 4957   Or wor 4958   [⊊] crpss 6834 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-rpss 6835 This theorem is referenced by:  sorpsscmpl  6846  enfin2i  9026  fin1a2lem13  9117
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