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Theorem brrpss 7272
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 7271 . 2 (𝐵 ∈ V → (𝐴 [] 𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴 [] 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2050  Vcvv 3415  wpss 3832   class class class wbr 4930   [] crpss 7268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931  df-opab 4993  df-xp 5414  df-rel 5415  df-rpss 7269
This theorem is referenced by:  porpss  7273  sorpss  7274  fin23lem40  9573  compssiso  9596  isfin1-3  9608  fin12  9635  zorng  9726  fin2solem  34319  psshepw  39497
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