Theorem ssnelpssd 4003

Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4002. (Contributed by David Moews, 1-May-2017.)

Hypotheses

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

Assertion

Ref	Expression
ssnelpssd	⊢ (𝜑 → 𝐴 ⊊ 𝐵)

Proof of Theorem ssnelpssd

Step	Hyp	Ref	Expression
1		ssnelpssd.2	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
2		ssnelpssd.3	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
3		ssnelpssd.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
4		ssnelpss 4002	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵))
6	1, 2, 5	mp2and 699	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2114   ⊆ wss 3843   ⊊ wpss 3844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-cleq 2730  df-clel 2811  df-ne 2935  df-pss 3862

This theorem is referenced by:  canth4  10147  mrieqv2d  17013  symgpssefmnd  18642  symggen  18716  pgpfac1lem1  19315  pgpfaclem2  19323