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Mirrors > Home > MPE Home > Th. List > ne0ii | Structured version Visualization version GIF version |
Description: If a class has elements, then it is nonempty. Inference associated with ne0i 4347. (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 4347 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐵 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ≠ 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-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-dif 3966 df-nul 4340 |
This theorem is referenced by: vn0ALT 4352 prnz 4782 tpnz 4784 pwne0 5363 onn0 6451 oawordeulem 8591 noinfep 9698 fin23lem31 10381 isfin1-3 10424 omina 10729 nnunb 12520 rpnnen1lem4 13020 rpnnen1lem5 13021 rexfiuz 15383 caurcvg 15710 caurcvg2 15711 caucvg 15712 infcvgaux1i 15890 divalglem2 16429 pc2dvds 16913 vdwmc2 17013 cnsubglem 21451 cnmsubglem 21466 pzriprnglem4 21513 pmatcollpw3 22806 zfbas 23920 nrginvrcn 24729 lebnumlem3 25009 caun0 25329 cnflduss 25404 cnfldcusp 25405 reust 25429 recusp 25430 nulmbl2 25585 itg2seq 25792 itg2monolem1 25800 c1lip1 26051 aannenlem2 26386 logbmpt 26846 tgcgr4 28554 shintcl 31359 chintcl 31361 nmoprepnf 31896 nmfnrepnf 31909 nmcexi 32055 snct 32731 esum0 34030 esumpcvgval 34059 bnj906 34923 satf0 35357 fmla1 35372 prv0 35415 bj-tagn0 36962 taupi 37306 ismblfin 37648 volsupnfl 37652 itg2addnclem 37658 ftc1anc 37688 incsequz 37735 isbnd3 37771 ssbnd 37775 onexomgt 43230 dflim5 43319 corclrcl 43697 imo72b2lem2 44157 imo72b2lem1 44159 imo72b2 44162 amgm2d 44188 nnn0 45328 ren0 45352 ioodvbdlimc1 45889 ioodvbdlimc2 45891 stirlinglem13 46042 fourierdlem103 46165 fourierdlem104 46166 fouriersw 46187 hoicvr 46504 2zlidl 48084 |
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