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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 2088 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
2 | 1 | exbii 1552 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑) |
3 | nfab1 2242 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 3 | n0rf 3322 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} → {𝑥 ∣ 𝜑} ≠ ∅) |
5 | 2, 4 | sylbir 134 | 1 ⊢ (∃𝑥𝜑 → {𝑥 ∣ 𝜑} ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1436 ∈ wcel 1448 {cab 2086 ≠ wne 2267 ∅c0 3310 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-v 2643 df-dif 3023 df-nul 3311 |
This theorem is referenced by: rabn0r 3336 |
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