Theorem brrpss 7709

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 7708	. 2 ⊢ (𝐵 ∈ V → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2109  Vcvv 3455   ⊊ wpss 3923   class class class wbr 5115   [⊊] crpss 7705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-br 5116  df-opab 5178  df-xp 5652  df-rel 5653  df-rpss 7706

This theorem is referenced by:  porpss  7710  sorpss  7711  fin23lem40  10322  compssiso  10345  isfin1-3  10357  fin12  10384  zorng  10475  fin2solem  37597  psshepw  43749