Theorem brrpssg

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

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

Proof

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