Theorem ssnelpssd 3615

Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3614. (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 3614	. . 3 ⊢ (A ⊆ B → ((C ∈ B ∧ ¬ C ∈ A) → A ⊊ B))
5	3, 4	syl 15	. 2 ⊢ (φ → ((C ∈ B ∧ ¬ C ∈ A) → A ⊊ B))
6	1, 2, 5	mp2and 660	1 ⊢ (φ → A ⊊ B)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∈ wcel 1710   ⊆ wss 3258   ⊊ wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-ne 2519  df-pss 3262

This theorem is referenced by: (None)