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Theorem brrpssg 6582
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 3118 . . 3
2 relrpss 6581 . . . 4
32brrelexi 5045 . . 3
41, 3anim12i 566 . 2
51adantr 465 . . 3
6 pssss 3598 . . . 4
7 ssexg 4598 . . . 4
86, 1, 7syl2anr 478 . . 3
95, 8jca 532 . 2
10 psseq1 3590 . . . 4
11 psseq2 3591 . . . 4
12 df-rpss 6580 . . . 4
1310, 11, 12brabg 4771 . . 3
1413ancoms 453 . 2
154, 9, 14pm5.21nd 900 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818   cvv 3109  C_wss 3475  C.wpss 3476   class class class wbr 4452   crpss 6579
This theorem is referenced by:  brrpss  6583  sorpssi  6586
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-xp 5010  df-rel 5011  df-rpss 6580
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