![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > nnn0 | Structured version Visualization version GIF version |
Description: The set of positive integers is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
nnn0 | ⊢ ℕ ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 11231 | . 2 ⊢ 1 ∈ ℕ | |
2 | 1 | ne0ii 4071 | 1 ⊢ ℕ ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2943 ∅c0 4063 1c1 10137 ℕcn 11220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-1cn 10194 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-om 7211 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-nn 11221 |
This theorem is referenced by: iocborel 41084 iunhoiioo 41403 iccvonmbllem 41405 preimageiingt 41443 preimaleiinlt 41444 salpreimagtge 41447 salpreimaltle 41448 smflimlem1 41492 |
Copyright terms: Public domain | W3C validator |