Theorem ssnelpss

Description: A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002)

	Ref	Expression
Assertion	ssnelpss	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴 ) → 𝐴 ⊊ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		nelneq2	⊢ ( ( 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴 ) → ¬ 𝐵 = 𝐴 )
2		eqcom	⊢ ( 𝐵 = 𝐴 ↔ 𝐴 = 𝐵 )
3	1 2	sylnib	⊢ ( ( 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴 ) → ¬ 𝐴 = 𝐵 )
4		dfpss2	⊢ ( 𝐴 ⊊ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵 ) )
5	4	baibr	⊢ ( 𝐴 ⊆ 𝐵 → ( ¬ 𝐴 = 𝐵 ↔ 𝐴 ⊊ 𝐵 ) )
6	3 5	syl5ib	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴 ) → 𝐴 ⊊ 𝐵 ) )