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Theorem ssnelpssd 3614
 Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3613. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ssnelpssd.1 (φA B)
ssnelpssd.2 (φC B)
ssnelpssd.3 (φ → ¬ C A)
Assertion
Ref Expression
ssnelpssd (φAB)

Proof of Theorem ssnelpssd
StepHypRef Expression
1 ssnelpssd.2 . 2 (φC B)
2 ssnelpssd.3 . 2 (φ → ¬ C A)
3 ssnelpssd.1 . . 3 (φA B)
4 ssnelpss 3613 . . 3 (A B → ((C B ¬ C A) → AB))
53, 4syl 15 . 2 (φ → ((C B ¬ C A) → AB))
61, 2, 5mp2and 660 1 (φAB)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∈ wcel 1710   ⊆ wss 3257   ⊊ wpss 3258 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 2518  df-pss 3261 This theorem is referenced by: (None)
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