Theorem brrpssg 7513

Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Assertion

Ref	Expression
brrpssg	⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))

Proof of Theorem brrpssg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3426	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2		relrpss 7512	. . . 4 ⊢ Rel [⊊]
3	2	brrelex1i 5605	. . 3 ⊢ (𝐴 [⊊] 𝐵 → 𝐴 ∈ V)
4	1, 3	anim12i 616	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 [⊊] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
5	1	adantr 484	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐵 ∈ V)
6		pssss 4010	. . . 4 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
7		ssexg 5216	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
8	6, 1, 7	syl2anr 600	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐴 ∈ V)
9	5, 8	jca 515	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
10		psseq1 4002	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦))
11		psseq2 4003	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵))
12		df-rpss 7511	. . . 4 ⊢ [⊊] = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦}
13	10, 11, 12	brabg 5420	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))
14	13	ancoms 462	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))
15	4, 9, 14	pm5.21nd 802	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2110  Vcvv 3408   ⊆ wss 3866   ⊊ wpss 3867   class class class wbr 5053   [⊊] crpss 7510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-rpss 7511

This theorem is referenced by:  brrpss  7514  sorpssi  7517