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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 512 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵)) |
4 | eleq1 2870 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)) | |
5 | eleq1 2870 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵)) | |
6 | 5 | notbid 319 | . . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑋 ∈ 𝐵)) |
7 | 4, 6 | anbi12d 630 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐵))) |
8 | 7 | spcegv 3540 | . . 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 3950 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
11 | 9, 10 | sylibr 235 | 1 ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵) |
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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