Theorem nssd 41364

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 514	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵))
4		eleq1 2900	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
5		eleq1 2900	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
6	5	notbid 320	. . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑋 ∈ 𝐵))
7	4, 6	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵)))
8	7	spcegv 3596	. . 3 ⊢ (𝑋 ∈ 𝐴 → ((𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵) → ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
9	1, 3, 8	sylc 65	. 2 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
10		nss 4028	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
11	9, 10	sylibr 236	1 ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ⊆ wss 3935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951

This theorem is referenced by: (None)