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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 9114 | . 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 1481 (class class class)co 5782 1c1 7645 + caddc 7647 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-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-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-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
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-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-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 |
This theorem is referenced by: elfzp1 9883 fznatpl1 9887 fzdifsuc 9892 fseq1p1m1 9905 flqge 10086 2tnp1ge0ge0 10105 ceiqm1l 10115 addmodlteq 10202 frec2uzzd 10204 frec2uzrdg 10213 uzsinds 10246 seq3f1olemqsumkj 10302 seq3f1olemqsumk 10303 bcp1nk 10540 bcval5 10541 hashfz 10599 resqrexlemdecn 10816 telfsumo 11267 fsumparts 11271 binomlem 11284 geo2sum 11315 cvgratnnlemseq 11327 cvgratnnlemabsle 11328 cvgratnnlemsumlt 11329 cvgratnnlemrate 11331 cvgratz 11333 mertenslemub 11335 mertenslemi1 11336 clim2prod 11340 clim2divap 11341 dvdsfac 11594 2tp1odd 11617 opoe 11628 zsupcllemstep 11674 prmind2 11837 hashdvds 11933 cvgcmp2nlemabs 13402 |
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