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Theorem sorpss 6585
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
Distinct variable group:   , ,

Proof of Theorem sorpss
StepHypRef Expression
1 porpss 6584 . . 3
21biantrur 506 . 2
3 sspsstri 3605 . . . 4
4 vex 3112 . . . . . 6
54brrpss 6583 . . . . 5
6 biid 236 . . . . 5
7 vex 3112 . . . . . 6
87brrpss 6583 . . . . 5
95, 6, 83orbi123i 1186 . . . 4
103, 9bitr4i 252 . . 3
11102ralbii 2889 . 2
12 df-so 4806 . 2
132, 11, 123bitr4ri 278 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  \/wo 368  /\wa 369  \/w3o 972  A.wral 2807  C_wss 3475  C.wpss 3476   class class class wbr 4452  Powpo 4803  Orwor 4804   crpss 6579
This theorem is referenced by:  sorpsscmpl  6591  enfin2i  8722  fin1a2lem13  8813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-po 4805  df-so 4806  df-xp 5010  df-rel 5011  df-rpss 6580
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