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Mirrors > Home > MPE Home > Th. List > abn0 | Structured version Visualization version GIF version |
Description: Nonempty class abstraction. See also ab0 4335. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2811, ax-8 2109. (Revised by Gino Giotto, 30-Aug-2024.) |
Ref | Expression |
---|---|
abn0 | ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ab0 4335 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | |
2 | 1 | notbii 320 | . 2 ⊢ (¬ {𝑥 ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥 ¬ 𝜑) |
3 | df-ne 2941 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ¬ {𝑥 ∣ 𝜑} = ∅) | |
4 | df-ex 1783 | . 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
5 | 2, 3, 4 | 3bitr4i 303 | 1 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 {cab 2710 ≠ wne 2940 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-ne 2941 df-dif 3914 df-nul 4284 |
This theorem is referenced by: intexab 5297 iinexg 5299 relimasn 6037 inisegn0 6051 mapprc 8772 modom 9191 tz9.1c 9671 scott0 9827 scott0s 9829 cp 9832 karden 9836 acnrcl 9983 aceq3lem 10061 cff 10189 cff1 10199 cfss 10206 domtriomlem 10383 axdclem 10460 nqpr 10955 supadd 12128 supmul 12132 hashf1lem2 14361 hashf1 14362 mreiincl 17481 efgval 19504 efger 19505 birthdaylem3 26319 disjex 31556 disjexc 31557 mppsval 34223 mblfinlem3 36163 ismblfin 36165 itg2addnc 36178 sdclem1 36248 upbdrech 43626 |
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