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Mirrors > Home > ILE Home > Th. List > neg1z | GIF version |
Description: -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.) |
Ref | Expression |
---|---|
neg1z | ⊢ -1 ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 8961 | . 2 ⊢ 1 ∈ ℕ | |
2 | nnnegz 9287 | . 2 ⊢ (1 ∈ ℕ → -1 ∈ ℤ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -1 ∈ ℤ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 1c1 7843 -cneg 8160 ℕcn 8950 ℤcz 9284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-sub 8161 df-neg 8162 df-inn 8951 df-z 9285 |
This theorem is referenced by: modqnegd 10412 modsumfzodifsn 10429 m1expcl 10577 n2dvdsm1 11953 pythagtriplem4 12303 cosq34lt1 14748 wilthlem1 14875 lgslem2 14880 lgsval 14883 lgsfvalg 14884 lgsfcl2 14885 lgsval2lem 14889 lgsvalmod 14898 lgsdir2lem3 14909 lgsdir2lem4 14910 lgsdir 14914 lgsdi 14916 lgsne0 14917 lgseisenlem2 14929 m1lgs 14930 apdiff 15275 |
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