Theorem ssnelpssd 3719

Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3718. (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 3718	. . 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 715	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∈ wcel 1990   ⊆ wss 3574   ⊊ wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ne 2795  df-pss 3590

This theorem is referenced by:  isfin4-3  9137  canth4  9469  mrieqv2d  16299  symggen  17890  pgpfac1lem1  18473  pgpfaclem2  18481