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Mirrors > Home > MPE Home > Th. List > neq0 | Structured version Visualization version GIF version |
Description: A class is not empty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 21-Jun-1993.) Avoid ax-11 2158, ax-12 2175. (Revised by Gino Giotto, 28-Jun-2024.) |
Ref | Expression |
---|---|
neq0 | ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1788 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | eq0 4258 | . . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
3 | 1, 2 | xchbinxr 338 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ 𝐴 = ∅) |
4 | 3 | bicomi 227 | 1 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ∅c0 4237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-dif 3869 df-nul 4238 |
This theorem is referenced by: n0 4261 ralidmOLD 4427 falseral0 4431 snprc 4633 pwpw0 4726 sssn 4739 pwsnOLD 4812 uni0b 4847 disjor 5033 rnep 5796 isomin 7146 mpoxneldm 7954 mpoxopynvov0g 7956 mpoxopxnop0 7957 erdisj 8443 ixpprc 8600 sucdom2 8755 domunsn 8796 isinf 8891 nfielex 8903 enp1i 8909 xpfi 8942 scottex 9501 acndom 9665 axcclem 10071 axpowndlem3 10213 canthp1lem1 10266 isumltss 15412 pf1rcl 21265 ppttop 21904 ntreq0 21974 txindis 22531 txconn 22586 fmfnfm 22855 ptcmplem2 22950 ptcmplem3 22951 bddmulibl 24736 g0wlk0 27739 wwlksnndef 27989 strlem1 30331 disjorf 30637 ddemeas 31916 tgoldbachgt 32355 bnj1143 32483 prv1n 33106 pibt2 35325 poimirlem25 35539 poimirlem27 35541 ineleq 36223 eqvreldisj 36464 grucollcld 41551 fnchoice 42245 founiiun0 42401 nzerooringczr 45303 mo0sn 45834 map0cor 45855 |
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