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Mirrors > Home > MPE Home > Th. List > ne0ii | Structured version Visualization version GIF version |
Description: If a set has elements, then it is not empty. Inference associated with ne0i 3880. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
n0ii.1 | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
ne0ii | ⊢ 𝐵 ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0ii.1 | . 2 ⊢ 𝐴 ∈ 𝐵 | |
2 | ne0i 3880 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐵 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-nul 3875 |
This theorem is referenced by: vn0 3883 prnz 4253 tpnz 4256 pwne0 4761 onn0 5706 oawordeulem 7521 noinfep 8440 fin23lem31 9048 isfin1-3 9091 omina 9392 rpnnen1lem4 11693 rpnnen1lem5 11694 rpnnen1lem4OLD 11699 rpnnen1lem5OLD 11700 rexfiuz 13935 caurcvg 14255 caurcvg2 14256 caucvg 14257 infcvgaux1i 14428 divalglem2 14956 pc2dvds 15421 vdwmc2 15521 cnsubglem 19614 cnmsubglem 19628 pmatcollpw3 20408 zfbas 21510 nrginvrcn 22306 lebnumlem3 22570 caun0 22887 cnflduss 22960 cnfldcusp 22961 reust 22977 recusp 22978 nulmbl2 23111 itg2seq 23315 itg2monolem1 23323 c1lip1 23564 aannenlem2 23888 logbmpt 24326 tgcgr4 25226 shintcl 27573 chintcl 27575 nmoprepnf 28110 nmfnrepnf 28123 nmcexi 28269 snct 28874 esum0 29438 esumpcvgval 29467 bnj906 30254 bj-tagn0 32160 taupi 32346 ismblfin 32620 volsupnfl 32624 itg2addnclem 32631 ftc1anc 32663 incsequz 32714 isbnd3 32753 ssbnd 32757 corclrcl 37018 imo72b2lem0 37487 imo72b2lem2 37489 imo72b2lem1 37493 imo72b2 37497 amgm2d 37523 nnn0 38536 ioodvbdlimc1 38823 ioodvbdlimc2 38825 stirlinglem13 38979 fourierdlem103 39102 fourierdlem104 39103 fouriersw 39124 hoicvr 39438 2zlidl 41724 |
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