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Theorem sorpss 6902
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 6901 . . 3 [] Po 𝐴
21biantrur 527 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ ( [] Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥)))
3 sspsstri 3692 . . . 4 ((𝑥𝑦𝑦𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
4 vex 3192 . . . . . 6 𝑦 ∈ V
54brrpss 6900 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
6 biid 251 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
7 vex 3192 . . . . . 6 𝑥 ∈ V
87brrpss 6900 . . . . 5 (𝑦 [] 𝑥𝑦𝑥)
95, 6, 83orbi123i 1250 . . . 4 ((𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
103, 9bitr4i 267 . . 3 ((𝑥𝑦𝑦𝑥) ↔ (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
11102ralbii 2976 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
12 df-so 5001 . 2 ( [] Or 𝐴 ↔ ( [] Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥)))
132, 11, 123bitr4ri 293 1 ( [] Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wa 384  w3o 1035  wral 2907  wss 3559  wpss 3560   class class class wbr 4618   Po wpo 4998   Or wor 4999   [] crpss 6896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-rpss 6897
This theorem is referenced by:  sorpsscmpl  6908  enfin2i  9094  fin1a2lem13  9185
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