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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nssd | Structured version Visualization version GIF version |
Description: Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
nssd.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
nssd.2 | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
nssd | ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nssd.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
2 | nssd.2 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐵) | |
3 | 1, 2 | jca 511 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵)) |
4 | eleq1 2817 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)) | |
5 | eleq1 2817 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵)) | |
6 | 5 | notbid 318 | . . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑋 ∈ 𝐵)) |
7 | 4, 6 | anbi12d 631 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵))) |
8 | 7 | spcegv 3583 | . . 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 4042 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
11 | 9, 10 | sylibr 233 | 1 ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ⊆ wss 3945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-in 3952 df-ss 3962 |
This theorem is referenced by: (None) |
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