Theorem brrpssg 7086

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 3364	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2		relrpss 7085	. . . 4 ⊢ Rel [⊊]
3	2	brrelexi 5298	. . 3 ⊢ (𝐴 [⊊] 𝐵 → 𝐴 ∈ V)
4	1, 3	anim12i 600	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 [⊊] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
5	1	adantr 466	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐵 ∈ V)
6		pssss 3852	. . . 4 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
7		ssexg 4938	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
8	6, 1, 7	syl2anr 584	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐴 ∈ V)
9	5, 8	jca 501	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
10		psseq1 3844	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦))
11		psseq2 3845	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵))
12		df-rpss 7084	. . . 4 ⊢ [⊊] = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦}
13	10, 11, 12	brabg 5127	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))
14	13	ancoms 455	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))
15	4, 9, 14	pm5.21nd 803	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∈ wcel 2145  Vcvv 3351   ⊆ wss 3723   ⊊ wpss 3724   class class class wbr 4786   [⊊] crpss 7083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-rpss 7084

This theorem is referenced by:  brrpss  7087  sorpssi  7090