Theorem prssbd

Description: If a pair is a subset of a class, the second element of the pair is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025)

	Ref	Expression
Hypotheses	prssbd.1	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
	prssbd.2	⊢ ( 𝜑 → { 𝐴 , 𝐵 } ⊆ 𝐶 )
Assertion	prssbd	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		prssbd.1	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
2		prssbd.2	⊢ ( 𝜑 → { 𝐴 , 𝐵 } ⊆ 𝐶 )
3		simpr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → 𝐴 ∈ V )
4	1	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → 𝐵 ∈ 𝑉 )
5	2	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → { 𝐴 , 𝐵 } ⊆ 𝐶 )
6		prssg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ↔ { 𝐴 , 𝐵 } ⊆ 𝐶 ) )
7	6	biimpar	⊢ ( ( ( 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ) ∧ { 𝐴 , 𝐵 } ⊆ 𝐶 ) → ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) )
8	3 4 5 7	syl21anc	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) )
9	8	simprd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → 𝐵 ∈ 𝐶 )
10		prprc1	⊢ ( ¬ 𝐴 ∈ V → { 𝐴 , 𝐵 } = { 𝐵 } )
11	10	adantl	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ∈ V ) → { 𝐴 , 𝐵 } = { 𝐵 } )
12	2	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ∈ V ) → { 𝐴 , 𝐵 } ⊆ 𝐶 )
13	11 12	eqsstrrd	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ∈ V ) → { 𝐵 } ⊆ 𝐶 )
14		snssg	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ 𝐶 ↔ { 𝐵 } ⊆ 𝐶 ) )
15	14	biimpar	⊢ ( ( 𝐵 ∈ 𝑉 ∧ { 𝐵 } ⊆ 𝐶 ) → 𝐵 ∈ 𝐶 )
16	1 13 15	syl2an2r	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ∈ V ) → 𝐵 ∈ 𝐶 )
17	9 16	pm2.61dan	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )