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Theorem porpss 6579
 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 3609 . . . . 5
2 psstr 3613 . . . . 5
3 vex 3121 . . . . . . . 8
43brrpss 6578 . . . . . . 7 []
54notbii 296 . . . . . 6 []
6 vex 3121 . . . . . . . . 9
76brrpss 6578 . . . . . . . 8 []
8 vex 3121 . . . . . . . . 9
98brrpss 6578 . . . . . . . 8 []
107, 9anbi12i 697 . . . . . . 7 [] []
118brrpss 6578 . . . . . . 7 []
1210, 11imbi12i 326 . . . . . 6 [] [] []
135, 12anbi12i 697 . . . . 5 [] [] [] []
141, 2, 13mpbir2an 918 . . . 4 [] [] [] []
1514rgenw 2828 . . 3 [] [] [] []
1615rgen2w 2829 . 2 [] [] [] []
17 df-po 4806 . 2 [] [] [] [] []
1816, 17mpbir 209 1 []
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369  wral 2817   wpss 3482   class class class wbr 4453   wpo 4804   [] crpss 6574 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-po 4806  df-xp 5011  df-rel 5012  df-rpss 6575 This theorem is referenced by:  sorpss  6580  fin23lem40  8743  isfin1-3  8778  zorng  8896  fin2so  29967
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