Theorem ssnelpssd 4067

Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4066. (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 4066	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵))
5	3, 4	syl 18	. 2 ⊢ (𝜑 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵))
6	1, 2, 5	mp2and 712	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2145   ⊆ wss 3902   ⊊ wpss 3903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-clel 2837  df-ne 2958  df-pss 3922

This theorem is used by:  canth4  10660  mrieqv2d  17733  symgpssefmnd  19529  symggen  19603  pgpfac1lem1  20209  pgpfaclem2  20217  ssdifidlprm  21555