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Theorem nssd 41738
Description: Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
nssd.1 (𝜑𝑋𝐴)
nssd.2 (𝜑 → ¬ 𝑋𝐵)
Assertion
Ref Expression
nssd (𝜑 → ¬ 𝐴𝐵)

Proof of Theorem nssd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nssd.1 . . 3 (𝜑𝑋𝐴)
2 nssd.2 . . . 4 (𝜑 → ¬ 𝑋𝐵)
31, 2jca 515 . . 3 (𝜑 → (𝑋𝐴 ∧ ¬ 𝑋𝐵))
4 eleq1 2880 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
5 eleq1 2880 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
65notbid 321 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥𝐵 ↔ ¬ 𝑋𝐵))
74, 6anbi12d 633 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ (𝑋𝐴 ∧ ¬ 𝑋𝐵)))
87spcegv 3548 . . 3 (𝑋𝐴 → ((𝑋𝐴 ∧ ¬ 𝑋𝐵) → ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵)))
91, 3, 8sylc 65 . 2 (𝜑 → ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
10 nss 3980 . 2 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
119, 10sylibr 237 1 (𝜑 → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  wss 3884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901
This theorem is referenced by: (None)
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