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Theorem ssnelpssd 3681
 Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3680. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ssnelpssd.1 (𝜑𝐴𝐵)
ssnelpssd.2 (𝜑𝐶𝐵)
ssnelpssd.3 (𝜑 → ¬ 𝐶𝐴)
Assertion
Ref Expression
ssnelpssd (𝜑𝐴𝐵)

Proof of Theorem ssnelpssd
StepHypRef Expression
1 ssnelpssd.2 . 2 (𝜑𝐶𝐵)
2 ssnelpssd.3 . 2 (𝜑 → ¬ 𝐶𝐴)
3 ssnelpssd.1 . . 3 (𝜑𝐴𝐵)
4 ssnelpss 3680 . . 3 (𝐴𝐵 → ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → 𝐴𝐵))
53, 4syl 17 . 2 (𝜑 → ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → 𝐴𝐵))
61, 2, 5mp2and 711 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540   ⊊ wpss 3541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-ne 2782  df-pss 3556 This theorem is referenced by:  isfin4-3  9020  canth4  9348  mrieqv2d  16122  symggen  17713  pgpfac1lem1  18296  pgpfaclem2  18304
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