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Mirrors > Home > ILE Home > Th. List > 0zd | GIF version |
Description: Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0zd | ⊢ (𝜑 → 0 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 8751 | . 2 ⊢ 0 ∈ ℤ | |
2 | 1 | a1i 9 | 1 ⊢ (𝜑 → 0 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 0cc0 7340 ℤcz 8740 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-1re 7429 ax-addrcl 7432 ax-rnegex 7444 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-un 3003 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-br 3844 df-iota 4975 df-fv 5018 df-ov 5647 df-neg 7646 df-z 8741 |
This theorem is referenced by: fzctr 9532 fzosubel3 9595 frecfzennn 9821 frechashgf1o 9823 0tonninf 9833 1tonninf 9834 exp3val 9945 exp0 9947 bcval 10145 bccmpl 10150 ibcval5 10159 bcpasc 10162 bccl 10163 hashcl 10177 hashfiv01gt1 10178 hashfz1 10179 hashen 10180 fihashneq0 10191 omgadd 10198 fihashdom 10199 fnfz0hash 10225 ffzo0hash 10227 fzomaxdiflem 10533 fsumzcl 10783 fisum0diag 10822 fisum0diag2 10828 binomlem 10864 binom1dif 10868 isumnn0nn 10874 expcnvre 10884 explecnv 10886 pwm1geoserap1 10889 geolim 10892 geolim2 10893 geo2sum 10895 geoisum 10898 geoisumr 10899 mertenslemub 10915 mertenslemi1 10916 mertenslem2 10917 mertensabs 10918 eftcl 10931 efval 10938 eff 10940 efcvg 10943 efcvgfsum 10944 reefcl 10945 ege2le3 10948 efcj 10950 efaddlem 10951 eftlub 10967 effsumlt 10969 efgt1p2 10972 efgt1p 10973 eflegeo 10979 eirraplem 11051 dvdsmod 11128 gcdn0gt0 11234 gcdaddm 11240 bezoutlemle 11262 nn0seqcvgd 11288 ialginv 11294 ialgcvg 11295 ialgcvga 11298 ialgfx 11299 eucalgval2 11300 eucialgcvga 11305 eucialg 11306 lcmcllem 11314 lcmid 11327 mulgcddvds 11341 divgcdcoprmex 11349 cncongr1 11350 cncongr2 11351 phiprmpw 11463 |
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