Theorem brrpss 7742

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 7741	. 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 2144  Vcvv 3450   ⊊ wpss 4013   class class class wbr 5102   [⊊] crpss 7738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-rpss 7739

This theorem is used by:  porpss  7743  sorpss  7744  fin23lem40  10429  compssiso  10452  isfin1-3  10464  fin12  10491  zorng  10582  fin2solem  38530  psshepw  44788  numtowerdt  47916