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Theorem porpss 6584
 Description: Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
porpss

Proof of Theorem porpss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pssirr 3603 . . . . 5
2 psstr 3607 . . . . 5
3 vex 3112 . . . . . . . 8
43brrpss 6583 . . . . . . 7
54notbii 296 . . . . . 6
6 vex 3112 . . . . . . . . 9
76brrpss 6583 . . . . . . . 8
8 vex 3112 . . . . . . . . 9
98brrpss 6583 . . . . . . . 8
107, 9anbi12i 697 . . . . . . 7
118brrpss 6583 . . . . . . 7
1210, 11imbi12i 326 . . . . . 6
135, 12anbi12i 697 . . . . 5
141, 2, 13mpbir2an 920 . . . 4
1514rgenw 2818 . . 3
1615rgen2w 2819 . 2
17 df-po 4805 . 2
1816, 17mpbir 209 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wral 2807  C.wpss 3476   class class class wbr 4452  Powpo 4803   crpss 6579 This theorem is referenced by:  sorpss  6585  fin23lem40  8752  isfin1-3  8787  zorng  8905  fin2so  30040 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-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-xp 5010  df-rel 5011  df-rpss 6580
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