Theorem nssd 45714

Description: Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
nssd.1	⊢ (𝜑 → 𝑋 ∈ 𝐴)
nssd.2	⊢ (𝜑 → ¬ 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
nssd	⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵)

Proof of Theorem nssd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nssd.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
2		nssd.2	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐵)
3	1, 2	jca 520	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵))
4		eleq1 2857	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
5		eleq1 2857	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
6	5	notbid 321	. . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑋 ∈ 𝐵))
7	4, 6	anbi12d 643	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵)))
8	7	spcegv 3565	. . 3 ⊢ (𝑋 ∈ 𝐴 → ((𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵) → ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
9	1, 3, 8	sylc 66	. 2 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
10		nss 4009	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
11	9, 10	sylibr 237	1 ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ss 3930

This theorem is referenced by: (None)