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Mirrors > Home > MPE Home > Th. List > eq0 | Structured version Visualization version GIF version |
Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.) |
Ref | Expression |
---|---|
eq0 | ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2974 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | eq0f 4302 | 1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∀wal 1526 = wceq 1528 ∈ wcel 2105 ∅c0 4288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-dif 3936 df-nul 4289 |
This theorem is referenced by: nel0 4308 0el 4317 ssdif0 4320 difin0ss 4325 inssdif0 4326 ralf0 4453 disjiun 5044 0ex 5202 reldm0 5791 iresn0n0 5916 uzwo 12299 hashgt0elex 13750 hausdiag 22181 rnelfmlem 22488 prv0 32574 wzel 33008 knoppndv 33770 bj-ab0 34121 bj-nul 34243 bj-nuliota 34244 bj-nuliotaALT 34245 nninfnub 34907 prtlem14 35890 nrhmzr 44072 zrninitoringc 44270 |
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