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Theorem brrpss 7514
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 7513 . 2 (𝐵 ∈ V → (𝐴 [] 𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴 [] 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2110  Vcvv 3408  wpss 3867   class class class wbr 5053   [] crpss 7510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-rpss 7511
This theorem is referenced by:  porpss  7515  sorpss  7516  fin23lem40  9965  compssiso  9988  isfin1-3  10000  fin12  10027  zorng  10118  fin2solem  35500  psshepw  41073
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