Theorem brrpss 7711

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 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)

Proof of Theorem brrpss

Step	Hyp	Ref	Expression
1		brrpss.a	. 2 ⊢ 𝐵 ∈ V
2		brrpssg 7710	. 2 ⊢ (𝐵 ∈ V → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)

Syntax hints:   ↔ wb 208   ∈ wcel 2144  Vcvv 3456   ⊊ wpss 3907   class class class wbr 5102   [⊊] crpss 7707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-rpss 7708

This theorem is referenced by:  porpss  7712  sorpss  7713  fin23lem40  10310  compssiso  10333  isfin1-3  10345  fin12  10372  zorng  10463  fin2solem  38110  psshepw  44369