Theorem nsspssun

Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004)

	Ref	Expression
Assertion	nsspssun	⊢ ( ¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ ( 𝐴 ∪ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ssun2	⊢ 𝐵 ⊆ ( 𝐴 ∪ 𝐵 )
2	1	biantrur	⊢ ( ¬ ( 𝐴 ∪ 𝐵 ) ⊆ 𝐵 ↔ ( 𝐵 ⊆ ( 𝐴 ∪ 𝐵 ) ∧ ¬ ( 𝐴 ∪ 𝐵 ) ⊆ 𝐵 ) )
3		ssid	⊢ 𝐵 ⊆ 𝐵
4	3	biantru	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵 ) )
5		unss	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵 ) ↔ ( 𝐴 ∪ 𝐵 ) ⊆ 𝐵 )
6	4 5	bitri	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ 𝐵 ) ⊆ 𝐵 )
7	2 6	xchnxbir	⊢ ( ¬ 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ⊆ ( 𝐴 ∪ 𝐵 ) ∧ ¬ ( 𝐴 ∪ 𝐵 ) ⊆ 𝐵 ) )
8		dfpss3	⊢ ( 𝐵 ⊊ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝐵 ⊆ ( 𝐴 ∪ 𝐵 ) ∧ ¬ ( 𝐴 ∪ 𝐵 ) ⊆ 𝐵 ) )
9	7 8	bitr4i	⊢ ( ¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ ( 𝐴 ∪ 𝐵 ) )