Theorem sorpssi

Description: Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014)

	Ref	Expression
Assertion	sorpssi	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		solin	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 [⊊] 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 [⊊] 𝐵 ) )
2		elex	⊢ ( 𝐶 ∈ 𝐴 → 𝐶 ∈ V )
3	2	ad2antll	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → 𝐶 ∈ V )
4		brrpssg	⊢ ( 𝐶 ∈ V → ( 𝐵 [⊊] 𝐶 ↔ 𝐵 ⊊ 𝐶 ) )
5	3 4	syl	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 [⊊] 𝐶 ↔ 𝐵 ⊊ 𝐶 ) )
6		biidd	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 = 𝐶 ↔ 𝐵 = 𝐶 ) )
7		elex	⊢ ( 𝐵 ∈ 𝐴 → 𝐵 ∈ V )
8	7	ad2antrl	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → 𝐵 ∈ V )
9		brrpssg	⊢ ( 𝐵 ∈ V → ( 𝐶 [⊊] 𝐵 ↔ 𝐶 ⊊ 𝐵 ) )
10	8 9	syl	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐶 [⊊] 𝐵 ↔ 𝐶 ⊊ 𝐵 ) )
11	5 6 10	3orbi123d	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( ( 𝐵 [⊊] 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 [⊊] 𝐵 ) ↔ ( 𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵 ) ) )
12	1 11	mpbid	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵 ) )
13		sspsstri	⊢ ( ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ↔ ( 𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵 ) )
14	12 13	sylibr	⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) )