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Theorem ssnelpssd 4104
Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4103. (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 4103 . . 3 (𝐴𝐵 → ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → 𝐴𝐵))
53, 4syl 17 . 2 (𝜑 → ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → 𝐴𝐵))
61, 2, 5mp2and 697 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wcel 2098  wss 3939  wpss 3940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2717  df-clel 2802  df-ne 2931  df-pss 3959
This theorem is referenced by:  canth4  10670  mrieqv2d  17618  symgpssefmnd  19354  symggen  19429  pgpfac1lem1  20035  pgpfaclem2  20043
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