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Mirrors > Home > ILE Home > Th. List > 2z | GIF version |
Description: Two is an integer. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
2z | ⊢ 2 ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 8905 | . 2 ⊢ 2 ∈ ℕ | |
2 | 1 | nnzi 9099 | 1 ⊢ 2 ∈ ℤ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 2c2 8795 ℤcz 9078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-2 8803 df-z 9079 |
This theorem is referenced by: nn0n0n1ge2b 9154 nn0lt2 9156 nn0le2is012 9157 zadd2cl 9204 eluz4eluz2 9389 uzuzle23 9393 2eluzge1 9398 eluz2b1 9422 nn01to3 9436 nn0ge2m1nnALT 9437 ige2m1fz 9921 fzctr 9941 fzo0to2pr 10026 fzo0to42pr 10028 qbtwnre 10065 2tnp1ge0ge0 10105 flhalf 10106 m1modge3gt1 10175 q2txmodxeq0 10188 sq1 10417 expnass 10429 sqrecapd 10459 sqoddm1div8 10475 bcn2m1 10547 bcn2p1 10548 4bc2eq6 10552 resqrexlemcalc1 10818 resqrexlemnmsq 10821 resqrexlemcvg 10823 resqrexlemglsq 10826 resqrexlemga 10827 resqrexlemsqa 10828 efgt0 11427 tanval3ap 11457 cos01bnd 11501 cos01gt0 11505 egt2lt3 11522 zeo3 11601 odd2np1 11606 even2n 11607 oddm1even 11608 oddp1even 11609 oexpneg 11610 2tp1odd 11617 2teven 11620 evend2 11622 oddp1d2 11623 ltoddhalfle 11626 opoe 11628 omoe 11629 opeo 11630 omeo 11631 m1expo 11633 m1exp1 11634 nn0o1gt2 11638 nn0o 11640 z0even 11644 n2dvds1 11645 z2even 11647 n2dvds3 11648 z4even 11649 4dvdseven 11650 flodddiv4 11667 6gcd4e2 11719 3lcm2e6woprm 11803 isprm3 11835 prmind2 11837 dvdsnprmd 11842 prm2orodd 11843 2prm 11844 3prm 11845 oddprmge3 11851 divgcdodd 11857 pw2dvds 11880 sqrt2irraplemnn 11893 oddennn 11941 evenennn 11942 unennn 11946 exmidunben 11975 sincos6thpi 12971 rpcxpsqrtth 13058 2logb9irr 13096 2logb9irrALT 13099 sqrt2cxp2logb9e3 13100 2logb9irrap 13102 ex-fl 13108 ex-dvds 13113 cvgcmp2nlemabs 13402 trilpolemlt1 13409 apdifflemr 13415 apdiff 13416 |
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