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Mirrors > Home > ILE Home > Th. List > ne0d | GIF version |
Description: Deduction form of ne0i 3453. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
ne0d.1 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
Ref | Expression |
---|---|
ne0d | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0d.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
2 | ne0i 3453 | . 2 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 ≠ wne 2364 ∅c0 3446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-v 2762 df-dif 3155 df-nul 3447 |
This theorem is referenced by: fihashelne0d 10855 mndbn0 12999 grpbn0 13089 bln0 14557 |
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