Theorem nssd 46119

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 521	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵))
4		eleq1 2849	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
5		eleq1 2849	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
6	5	notbid 321	. . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑋 ∈ 𝐵))
7	4, 6	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵)))
8	7	spcegv 3552	. . 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 3995	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
11	9, 10	sylibr 237	1 ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ⊆ wss 3899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ss 3916

This theorem is used by: (None)