Theorem brrpss 7730

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 7729	. 2 ⊢ (𝐵 ∈ V → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  Vcvv 3453   ⊊ wpss 3903   class class class wbr 5107   [⊊] crpss 7726

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-rpss 7727

This theorem is used by:  porpss  7731  sorpss  7732  fin23lem40  10356  compssiso  10379  isfin1-3  10391  fin12  10418  zorng  10509  fin2solem  38347  psshepw  44615  numtowerdt  47721