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Theorem brrpssg 7744
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 3499 . . 3 (𝐵𝑉𝐵 ∈ V)
2 relrpss 7743 . . . 4 Rel []
32brrelex1i 5745 . . 3 (𝐴 [] 𝐵𝐴 ∈ V)
41, 3anim12i 613 . 2 ((𝐵𝑉𝐴 [] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
51adantr 480 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐵 ∈ V)
6 pssss 4108 . . . 4 (𝐴𝐵𝐴𝐵)
7 ssexg 5329 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
86, 1, 7syl2anr 597 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐴 ∈ V)
95, 8jca 511 . 2 ((𝐵𝑉𝐴𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
10 psseq1 4100 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 psseq2 4101 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-rpss 7742 . . . 4 [] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5549 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
1413ancoms 458 . 2 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
154, 9, 14pm5.21nd 802 1 (𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  Vcvv 3478  wss 3963  wpss 3964   class class class wbr 5148   [] crpss 7741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-rpss 7742
This theorem is referenced by:  brrpss  7745  sorpssi  7748
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