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Theorem nssd 43116
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 513 . . 3 (𝜑 → (𝑋𝐴 ∧ ¬ 𝑋𝐵))
4 eleq1 2826 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
5 eleq1 2826 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
65notbid 318 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥𝐵 ↔ ¬ 𝑋𝐵))
74, 6anbi12d 632 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ (𝑋𝐴 ∧ ¬ 𝑋𝐵)))
87spcegv 3555 . . 3 (𝑋𝐴 → ((𝑋𝐴 ∧ ¬ 𝑋𝐵) → ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵)))
91, 3, 8sylc 65 . 2 (𝜑 → ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
10 nss 4005 . 2 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
119, 10sylibr 233 1 (𝜑 → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wss 3909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3446  df-in 3916  df-ss 3926
This theorem is referenced by: (None)
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