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Theorem brrpss 7166
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 7165 . 2 (𝐵 ∈ V → (𝐴 [] 𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴 [] 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wcel 2156  Vcvv 3391  wpss 3770   class class class wbr 4844   [] crpss 7162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-xp 5317  df-rel 5318  df-rpss 7163
This theorem is referenced by:  porpss  7167  sorpss  7168  fin23lem40  9454  compssiso  9477  isfin1-3  9489  fin12  9516  zorng  9607  fin2solem  33706  psshepw  38579
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