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Theorem brrpss 7087
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 7086 . 2 (𝐵 ∈ V → (𝐴 [] 𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴 [] 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2145  Vcvv 3351  wpss 3724   class class class wbr 4786   [] crpss 7083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-rpss 7084
This theorem is referenced by:  porpss  7088  sorpss  7089  fin23lem40  9375  compssiso  9398  isfin1-3  9410  fin12  9437  zorng  9528  fin2solem  33728  psshepw  38608
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