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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 4190 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 2961 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | eq0f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | neq0f 4186 | . 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
4 | 1, 3 | bitri 267 | 1 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 = wceq 1508 ∃wex 1743 ∈ wcel 2051 Ⅎwnfc 2909 ≠ wne 2960 ∅c0 4172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-11 2094 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-dif 3825 df-nul 4173 |
This theorem is referenced by: n0 4190 abn0 4216 cp 9112 ordtconnlem1 30843 inn0f 40792 |
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