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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 4359 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 2939 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | eq0f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | neq0f 4354 | . 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 1537 ∃wex 1776 ∈ wcel 2106 Ⅎwnfc 2888 ≠ 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-8 2108 ax-9 2116 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-clel 2814 df-nfc 2890 df-ne 2939 df-dif 3966 df-nul 4340 |
This theorem is referenced by: inn0f 4377 cp 9929 ordtconnlem1 33885 iinss2d 45100 r19.3rzf 45101 |
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