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Theorem sorpss 7089
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 7088 . . 3 [] Po 𝐴
21biantrur 514 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ ( [] Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥)))
3 sspsstri 3859 . . . 4 ((𝑥𝑦𝑦𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
4 vex 3354 . . . . . 6 𝑦 ∈ V
54brrpss 7087 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
6 biid 251 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
7 vex 3354 . . . . . 6 𝑥 ∈ V
87brrpss 7087 . . . . 5 (𝑦 [] 𝑥𝑦𝑥)
95, 6, 83orbi123i 1159 . . . 4 ((𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
103, 9bitr4i 267 . . 3 ((𝑥𝑦𝑦𝑥) ↔ (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
11102ralbii 3130 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 [] 𝑦𝑥 = 𝑦𝑦 [] 𝑥))
12 df-so 5171 . 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  wa 382  wo 826  w3o 1070  wral 3061  wss 3723  wpss 3724   class class class wbr 4786   Po wpo 5168   Or wor 5169   [] crpss 7083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-rpss 7084
This theorem is referenced by:  sorpsscmpl  7095  enfin2i  9345  fin1a2lem13  9436
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