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Theorem sorpss 7274
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 7273 . . 3 [] Po 𝐴
21biantrur 523 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ ( [] Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥)))
3 sspsstri 3971 . . . 4 ((𝑥𝑦𝑦𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
4 vex 3418 . . . . . 6 𝑦 ∈ V
54brrpss 7272 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
6 biid 253 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
7 vex 3418 . . . . . 6 𝑥 ∈ V
87brrpss 7272 . . . . 5 (𝑦 [] 𝑥𝑦𝑥)
95, 6, 83orbi123i 1136 . . . 4 ((𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
103, 9bitr4i 270 . . 3 ((𝑥𝑦𝑦𝑥) ↔ (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
11102ralbii 3116 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
12 df-so 5328 . 2 ( [] Or 𝐴 ↔ ( [] Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥)))
132, 11, 123bitr4ri 296 1 ( [] Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wo 833  w3o 1067  wral 3088  wss 3831  wpss 3832   class class class wbr 4930   Po wpo 5325   Or wor 5326   [] crpss 7268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931  df-opab 4993  df-po 5327  df-so 5328  df-xp 5414  df-rel 5415  df-rpss 7269
This theorem is referenced by:  sorpsscmpl  7280  enfin2i  9543  fin1a2lem13  9634
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