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Theorem brrpssg 6526
 Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
brrpssg []

Proof of Theorem brrpssg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2966 . . 3
2 relrpss 6525 . . . 4 []
32brrelexi 4920 . . 3 []
41, 3anim12i 551 . 2 []
51adantr 453 . . 3
6 pssss 3444 . . . 4
7 ssexg 4351 . . . 4
86, 1, 7syl2anr 466 . . 3
95, 8jca 520 . 2
10 psseq1 3436 . . . 4
11 psseq2 3437 . . . 4
12 df-rpss 6524 . . . 4 []
1310, 11, 12brabg 4476 . . 3 []
1413ancoms 441 . 2 []
154, 9, 14pm5.21nd 870 1 []
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wcel 1726  cvv 2958   wss 3322   wpss 3323   class class class wbr 4214   [] crpss 6523 This theorem is referenced by:  brrpss  6527  sorpssi  6530 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-rpss 6524
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