Theorem brrpss 7669

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 7668	. 2 ⊢ (𝐵 ∈ V → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3427   ⊊ wpss 3886   class class class wbr 5074   [⊊] crpss 7665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-rel 5627  df-rpss 7666

This theorem is referenced by:  porpss  7670  sorpss  7671  fin23lem40  10262  compssiso  10285  isfin1-3  10297  fin12  10324  zorng  10415  fin2solem  37915  psshepw  44203