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Theorem brrpssg 7736
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 3492 . . 3 (𝐵𝑉𝐵 ∈ V)
2 relrpss 7735 . . . 4 Rel []
32brrelex1i 5738 . . 3 (𝐴 [] 𝐵𝐴 ∈ V)
41, 3anim12i 611 . 2 ((𝐵𝑉𝐴 [] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
51adantr 479 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐵 ∈ V)
6 pssss 4095 . . . 4 (𝐴𝐵𝐴𝐵)
7 ssexg 5327 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
86, 1, 7syl2anr 595 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐴 ∈ V)
95, 8jca 510 . 2 ((𝐵𝑉𝐴𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
10 psseq1 4087 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
11 psseq2 4088 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
12 df-rpss 7734 . . . 4 [] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1310, 11, 12brabg 5545 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
1413ancoms 457 . 2 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [] 𝐵𝐴𝐵))
154, 9, 14pm5.21nd 800 1 (𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  Vcvv 3473  wss 3949  wpss 3950   class class class wbr 5152   [] crpss 7733
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-rpss 7734
This theorem is referenced by:  brrpss  7737  sorpssi  7740
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