Theorem ssnelpssd 4088

Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4087. (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 4087	. . 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 697	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∈ wcel 2110   ⊆ wss 3935   ⊊ wpss 3936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-clel 2893  df-ne 3017  df-pss 3953

This theorem is referenced by:  canth4  10063  mrieqv2d  16904  symgpssefmnd  18518  symggen  18592  pgpfac1lem1  19190  pgpfaclem2  19198