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Mirrors > Home > ILE Home > Th. List > peano2zd | GIF version |
Description: Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
zred.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
Ref | Expression |
---|---|
peano2zd | ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | peano2z 9291 | . 2 ⊢ (𝐴 ∈ ℤ → (𝐴 + 1) ∈ ℤ) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 (class class class)co 5877 1c1 7814 + caddc 7816 ℤcz 9255 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-sub 8132 df-neg 8133 df-inn 8922 df-n0 9179 df-z 9256 |
This theorem is referenced by: elfzp1 10074 fznatpl1 10078 fzdifsuc 10083 fseq1p1m1 10096 flqge 10284 2tnp1ge0ge0 10303 ceiqm1l 10313 addmodlteq 10400 frec2uzzd 10402 frec2uzrdg 10411 uzsinds 10444 seq3f1olemqsumkj 10500 seq3f1olemqsumk 10501 bcp1nk 10744 bcval5 10745 hashfz 10803 resqrexlemdecn 11023 telfsumo 11476 fsumparts 11480 binomlem 11493 geo2sum 11524 cvgratnnlemseq 11536 cvgratnnlemabsle 11537 cvgratnnlemsumlt 11538 cvgratnnlemrate 11540 cvgratz 11542 mertenslemub 11544 mertenslemi1 11545 clim2prod 11549 clim2divap 11550 fprodntrivap 11594 fprodeq0 11627 dvdsfac 11868 2tp1odd 11891 opoe 11902 zsupcllemstep 11948 suprzubdc 11955 prmind2 12122 hashdvds 12223 eulerthlemrprm 12231 pcprendvds 12292 nninfdclemcl 12451 nninfdclemp1 12453 lgslem1 14486 lgsval 14490 lgsfvalg 14491 lgsval2lem 14496 lgsvalmod 14505 lgseisenlem1 14535 m1lgs 14537 cvgcmp2nlemabs 14865 |
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