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Mirrors > Home > MPE Home > Th. List > abn0 | Structured version Visualization version GIF version |
Description: Nonempty class abstraction. See also ab0 4373. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2810, ax-8 2108. (Revised by Gino Giotto, 30-Aug-2024.) |
Ref | Expression |
---|---|
abn0 | ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ab0 4373 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | |
2 | 1 | notbii 319 | . 2 ⊢ (¬ {𝑥 ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥 ¬ 𝜑) |
3 | df-ne 2941 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ¬ {𝑥 ∣ 𝜑} = ∅) | |
4 | df-ex 1782 | . 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
5 | 2, 3, 4 | 3bitr4i 302 | 1 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1539 = wceq 1541 ∃wex 1781 {cab 2709 ≠ wne 2940 ∅c0 4321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-ne 2941 df-dif 3950 df-nul 4322 |
This theorem is referenced by: intexab 5338 iinexg 5340 relimasn 6080 inisegn0 6094 mapprc 8820 modom 9240 tz9.1c 9721 scott0 9877 scott0s 9879 cp 9882 karden 9886 acnrcl 10033 aceq3lem 10111 cff 10239 cff1 10249 cfss 10256 domtriomlem 10433 axdclem 10510 nqpr 11005 supadd 12178 supmul 12182 hashf1lem2 14413 hashf1 14414 mreiincl 17536 efgval 19579 efger 19580 birthdaylem3 26447 disjex 31810 disjexc 31811 mppsval 34551 mblfinlem3 36515 ismblfin 36517 itg2addnc 36530 sdclem1 36599 upbdrech 44001 |
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