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Theorem nssd 38317
 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 553 . . 3 (𝜑 → (𝑋𝐴 ∧ ¬ 𝑋𝐵))
4 eleq1 2676 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
5 eleq1 2676 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
65notbid 307 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥𝐵 ↔ ¬ 𝑋𝐵))
74, 6anbi12d 743 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ (𝑋𝐴 ∧ ¬ 𝑋𝐵)))
87spcegv 3267 . . 3 (𝑋𝐴 → ((𝑋𝐴 ∧ ¬ 𝑋𝐵) → ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵)))
91, 3, 8sylc 63 . 2 (𝜑 → ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
10 nss 3626 . 2 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
119, 10sylibr 223 1 (𝜑 → ¬ 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ⊆ wss 3540 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-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
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