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Mirrors > Home > NFE Home > Th. List > abn0 | GIF version |
Description: Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) |
Ref | Expression |
---|---|
abn0 | ⊢ ({x ∣ φ} ≠ ∅ ↔ ∃xφ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfab1 2491 | . . 3 ⊢ Ⅎx{x ∣ φ} | |
2 | 1 | n0f 3558 | . 2 ⊢ ({x ∣ φ} ≠ ∅ ↔ ∃x x ∈ {x ∣ φ}) |
3 | abid 2341 | . . 3 ⊢ (x ∈ {x ∣ φ} ↔ φ) | |
4 | 3 | exbii 1582 | . 2 ⊢ (∃x x ∈ {x ∣ φ} ↔ ∃xφ) |
5 | 2, 4 | bitri 240 | 1 ⊢ ({x ∣ φ} ≠ ∅ ↔ ∃xφ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wex 1541 ∈ wcel 1710 {cab 2339 ≠ wne 2516 ∅c0 3550 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: ab0 3569 rabn0 3570 imasn 5018 frds 5935 mapprc 6004 map0b 6024 map0 6025 |
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