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| 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 4352 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 2940 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
| 2 | eq0f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | neq0f 4347 | . 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | 
| 4 | 1, 3 | bitri 275 | 1 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1539 ∃wex 1778 ∈ wcel 2107 Ⅎwnfc 2889 ≠ wne 2939 ∅c0 4332 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-dif 3953 df-nul 4333 | 
| This theorem is referenced by: inn0f 4370 cp 9932 ordtconnlem1 33924 iinss2d 45167 r19.3rzf 45168 | 
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