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| Mirrors > Home > MPE Home > Th. List > Mathboxes > n0limd | Structured version Visualization version GIF version | ||
| Description: Deduction rule for nonempty classes. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| Ref | Expression |
|---|---|
| n0limd.1 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
| n0limd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
| Ref | Expression |
|---|---|
| n0limd | ⊢ (𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0limd.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
| 2 | n0 4353 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
| 3 | 1, 2 | sylib 218 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
| 4 | n0limd.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
| 5 | 3, 4 | exlimddv 1935 | 1 ⊢ (𝜑 → 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-ne 2941 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: dimlssid 33683 fldextrspunlem1 33725 |
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