Theorem ssnelpssd

Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss . (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	ssnelpssd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	ssnelpssd.2	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )
	ssnelpssd.3	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐴 )
Assertion	ssnelpssd	⊢ ( 𝜑 → 𝐴 ⊊ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ssnelpssd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		ssnelpssd.2	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )
3		ssnelpssd.3	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐴 )
4		ssnelpss	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴 ) → 𝐴 ⊊ 𝐵 ) )
5	1 4	syl	⊢ ( 𝜑 → ( ( 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴 ) → 𝐴 ⊊ 𝐵 ) )
6	2 3 5	mp2and	⊢ ( 𝜑 → 𝐴 ⊊ 𝐵 )