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| Mirrors > Home > MPE Home > Th. List > nelrdva | Structured version Visualization version GIF version | ||
| Description: Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) | 
| Ref | Expression | 
|---|---|
| nelrdva.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝐵) | 
| Ref | Expression | 
|---|---|
| nelrdva | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqidd 2737 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → 𝐵 = 𝐵) | |
| 2 | eleq1 2828 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
| 3 | 2 | anbi2d 630 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝐵 ∈ 𝐴))) | 
| 4 | neeq1 3002 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ≠ 𝐵 ↔ 𝐵 ≠ 𝐵)) | |
| 5 | 3, 4 | imbi12d 344 | . . . . 5 ⊢ (𝑥 = 𝐵 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝐵) ↔ ((𝜑 ∧ 𝐵 ∈ 𝐴) → 𝐵 ≠ 𝐵))) | 
| 6 | nelrdva.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝐵) | |
| 7 | 5, 6 | vtoclg 3553 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → ((𝜑 ∧ 𝐵 ∈ 𝐴) → 𝐵 ≠ 𝐵)) | 
| 8 | 7 | anabsi7 671 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → 𝐵 ≠ 𝐵) | 
| 9 | 8 | neneqd 2944 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 = 𝐵) | 
| 10 | 1, 9 | pm2.65da 816 | 1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 | 
| This theorem is referenced by: ustfilxp 24222 metustfbas 24571 drngmxidl 33506 fourierdlem72 46198 | 
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