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Theorem brrpssg 7667
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 (𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))

Proof of Theorem brrpssg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3466 . . 3 (𝐵𝑉𝐵 ∈ V)
2 relrpss 7666 . . . 4 Rel []
32brrelex1i 5693 . . 3 (𝐴 [] 𝐵𝐴 ∈ V)
41, 3anim12i 614 . 2 ((𝐵𝑉𝐴 [] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
51adantr 482 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐵 ∈ V)
6 pssss 4060 . . . 4 (𝐴𝐵𝐴𝐵)
7 ssexg 5285 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
86, 1, 7syl2anr 598 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐴 ∈ V)
95, 8jca 513 . 2 ((𝐵𝑉𝐴𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
10 psseq1 4052 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 psseq2 4053 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-rpss 7665 . . . 4 [] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5501 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
1413ancoms 460 . 2 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
154, 9, 14pm5.21nd 801 1 (𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  Vcvv 3448  wss 3915  wpss 3916   class class class wbr 5110   [] crpss 7664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-rpss 7665
This theorem is referenced by:  brrpss  7668  sorpssi  7671
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