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Mirrors > Home > MPE Home > Th. List > abn0 | Structured version Visualization version GIF version |
Description: Nonempty class abstraction. See also ab0 4386. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2814, ax-8 2108. (Revised by GG, 30-Aug-2024.) |
Ref | Expression |
---|---|
abn0 | ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ab0 4386 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | |
2 | 1 | notbii 320 | . 2 ⊢ (¬ {𝑥 ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥 ¬ 𝜑) |
3 | df-ne 2939 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ¬ {𝑥 ∣ 𝜑} = ∅) | |
4 | df-ex 1777 | . 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 206 ∀wal 1535 = wceq 1537 ∃wex 1776 {cab 2712 ≠ wne 2938 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-ne 2939 df-dif 3966 df-nul 4340 |
This theorem is referenced by: intexab 5352 iinexg 5354 relimasn 6105 inisegn0 6119 mapprc 8869 modom 9278 tz9.1c 9768 scott0 9924 scott0s 9926 cp 9929 karden 9933 acnrcl 10080 aceq3lem 10158 cff 10286 cff1 10296 cfss 10303 domtriomlem 10480 axdclem 10557 nqpr 11052 supadd 12234 supmul 12238 hashf1lem2 14492 hashf1 14493 mreiincl 17641 efgval 19750 efger 19751 birthdaylem3 27011 disjex 32612 disjexc 32613 mppsval 35557 mblfinlem3 37646 ismblfin 37648 itg2addnc 37661 sdclem1 37730 upbdrech 45256 |
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