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Mirrors > Home > MPE Home > Th. List > abn0 | Structured version Visualization version GIF version |
Description: Nonempty class abstraction. See also ab0 4375. (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 4375 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | |
2 | 1 | notbii 320 | . 2 ⊢ (¬ {𝑥 ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥 ¬ 𝜑) |
3 | df-ne 2942 | . 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 2941 ∅c0 4323 |
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 2942 df-dif 3952 df-nul 4324 |
This theorem is referenced by: intexab 5340 iinexg 5342 relimasn 6084 inisegn0 6098 mapprc 8824 modom 9244 tz9.1c 9725 scott0 9881 scott0s 9883 cp 9886 karden 9890 acnrcl 10037 aceq3lem 10115 cff 10243 cff1 10253 cfss 10260 domtriomlem 10437 axdclem 10514 nqpr 11009 supadd 12182 supmul 12186 hashf1lem2 14417 hashf1 14418 mreiincl 17540 efgval 19585 efger 19586 birthdaylem3 26458 disjex 31823 disjexc 31824 mppsval 34563 mblfinlem3 36527 ismblfin 36529 itg2addnc 36542 sdclem1 36611 upbdrech 44015 |
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