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Theorem nssd 40911
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 512 . . 3 (𝜑 → (𝑋𝐴 ∧ ¬ 𝑋𝐵))
4 eleq1 2870 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
5 eleq1 2870 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
65notbid 319 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥𝐵 ↔ ¬ 𝑋𝐵))
74, 6anbi12d 630 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ (𝑋𝐴 ∧ ¬ 𝑋𝐵)))
87spcegv 3540 . . 3 (𝑋𝐴 → ((𝑋𝐴 ∧ ¬ 𝑋𝐵) → ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵)))
91, 3, 8sylc 65 . 2 (𝜑 → ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
10 nss 3950 . 2 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
119, 10sylibr 235 1 (𝜑 → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1522  wex 1761  wcel 2081  wss 3859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-in 3866  df-ss 3874
This theorem is referenced by: (None)
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