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Mirrors > Home > ILE Home > Th. List > peano2uz | Unicode version |
Description: Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.) |
Ref | Expression |
---|---|
peano2uz |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 999 |
. . 3
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2 | peano2z 9318 |
. . . 4
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3 | 2 | 3ad2ant2 1021 |
. . 3
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4 | zre 9286 |
. . . 4
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5 | zre 9286 |
. . . . 5
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6 | letrp1 8834 |
. . . . 5
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7 | 5, 6 | syl3an2 1283 |
. . . 4
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8 | 4, 7 | syl3an1 1282 |
. . 3
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9 | 1, 3, 8 | 3jca 1179 |
. 2
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10 | eluz2 9563 |
. 2
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11 | eluz2 9563 |
. 2
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12 | 9, 10, 11 | 3imtr4i 201 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-addass 7942 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-ltadd 7956 |
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-nel 2456 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-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-inn 8949 df-n0 9206 df-z 9283 df-uz 9558 |
This theorem is referenced by: peano2uzs 9613 peano2uzr 9614 uzaddcl 9615 fzsplit 10080 fzssp1 10096 fzsuc 10098 fzpred 10099 fzp1ss 10102 fzp1elp1 10104 fztp 10107 fzneuz 10130 fzosplitsnm1 10238 fzofzp1 10256 fzosplitsn 10262 fzostep1 10266 frec2uzuzd 10432 frecuzrdgrrn 10438 frec2uzrdg 10439 frecuzrdgrcl 10440 frecuzrdgsuc 10444 frecuzrdgrclt 10445 frecuzrdgg 10446 frecuzrdgsuctlem 10453 frecfzen2 10457 fzfig 10460 uzsinds 10472 iseqovex 10486 seq3val 10488 seqvalcd 10489 seqf 10491 seq3p1 10492 seq3split 10509 seq3homo 10540 seq3z 10541 ser3ge0 10547 faclbnd3 10754 bcm1k 10771 seq3coll 10853 clim2ser 11376 clim2ser2 11377 serf0 11391 fsump1 11459 fsump1i 11472 fsumparts 11509 isum1p 11531 cvgratnnlemmn 11564 mertenslemi1 11574 clim2prod 11578 clim2divap 11579 fprodntrivap 11623 fprodp1 11639 fprodabs 11655 zsupcllemstep 11977 infssuzex 11981 pcfac 12381 |
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