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Mirrors > Home > MPE Home > Th. List > n0f | Structured version Visualization version GIF version |
Description: A class is nonempty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 4350 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by NM, 17-Oct-2003.) |
Ref | Expression |
---|---|
eq0f.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
n0f | ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2938 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | eq0f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | neq0f 4345 | . 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
4 | 1, 3 | bitri 274 | 1 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Ⅎwnfc 2879 ≠ wne 2937 ∅c0 4326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-dif 3952 df-nul 4327 |
This theorem is referenced by: n0OLD 4353 abn0OLD 4385 cp 9922 ordtconnlem1 33558 inn0f 44468 iinss2d 44558 r19.3rzf 44559 |
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