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Theorem brrpss 6838
 Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypothesis
Ref Expression
brrpss.a 𝐵 ∈ V
Assertion
Ref Expression
brrpss (𝐴 [] 𝐵𝐴𝐵)

Proof of Theorem brrpss
StepHypRef Expression
1 brrpss.a . 2 𝐵 ∈ V
2 brrpssg 6837 . 2 (𝐵 ∈ V → (𝐴 [] 𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴 [] 𝐵𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  Vcvv 3173   ⊊ wpss 3541   class class class wbr 4583   [⊊] crpss 6834 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-rpss 6835 This theorem is referenced by:  porpss  6839  sorpss  6840  fin23lem40  9056  compssiso  9079  isfin1-3  9091  fin12  9118  zorng  9209  fin2solem  32565  psshepw  37102
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