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Mirrors > Home > ILE Home > Th. List > abn0r | GIF version |
Description: Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.) |
Ref | Expression |
---|---|
abn0r | ⊢ (∃𝑥𝜑 → {𝑥 ∣ 𝜑} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid 2069 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
2 | 1 | exbii 1536 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑) |
3 | nfab1 2221 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 3 | n0rf 3260 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} → {𝑥 ∣ 𝜑} ≠ ∅) |
5 | 2, 4 | sylbir 133 | 1 ⊢ (∃𝑥𝜑 → {𝑥 ∣ 𝜑} ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1421 ∈ wcel 1433 {cab 2067 ≠ wne 2245 ∅c0 3251 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-v 2603 df-dif 2975 df-nul 3252 |
This theorem is referenced by: rabn0r 3271 |
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