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Theorem brrpssg 7711
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 3487 . . 3 (𝐵𝑉𝐵 ∈ V)
2 relrpss 7710 . . . 4 Rel []
32brrelex1i 5725 . . 3 (𝐴 [] 𝐵𝐴 ∈ V)
41, 3anim12i 612 . 2 ((𝐵𝑉𝐴 [] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
51adantr 480 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐵 ∈ V)
6 pssss 4090 . . . 4 (𝐴𝐵𝐴𝐵)
7 ssexg 5316 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
86, 1, 7syl2anr 596 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐴 ∈ V)
95, 8jca 511 . 2 ((𝐵𝑉𝐴𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
10 psseq1 4082 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 psseq2 4083 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-rpss 7709 . . . 4 [] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5532 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
1413ancoms 458 . 2 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
154, 9, 14pm5.21nd 799 1 (𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2098  Vcvv 3468  wss 3943  wpss 3944   class class class wbr 5141   [] crpss 7708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-rpss 7709
This theorem is referenced by:  brrpss  7712  sorpssi  7715
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