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Mirrors > Home > ILE Home > Th. List > peano5nnnn | Unicode version |
Description: Peano's inductive postulate. This is a counterpart to peano5nni 8851 designed for real number axioms which involve natural numbers (notably, axcaucvg 7832). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nntopi.n |
Ref | Expression |
---|---|
peano5nnnn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5843 | . . . 4 | |
2 | 1 | eleq1d 2233 | . . 3 |
3 | 2 | cbvralv 2689 | . 2 |
4 | ax1re 7794 | . . . . 5 | |
5 | elin 3300 | . . . . . 6 | |
6 | 5 | biimpri 132 | . . . . 5 |
7 | 4, 6 | mpan2 422 | . . . 4 |
8 | inss1 3337 | . . . . . 6 | |
9 | ssralv 3201 | . . . . . 6 | |
10 | 8, 9 | ax-mp 5 | . . . . 5 |
11 | inss2 3338 | . . . . . . . 8 | |
12 | 11 | sseli 3133 | . . . . . . 7 |
13 | axaddrcl 7797 | . . . . . . . 8 | |
14 | 4, 13 | mpan2 422 | . . . . . . 7 |
15 | elin 3300 | . . . . . . . 8 | |
16 | 15 | simplbi2com 1431 | . . . . . . 7 |
17 | 12, 14, 16 | 3syl 17 | . . . . . 6 |
18 | 17 | ralimia 2525 | . . . . 5 |
19 | 10, 18 | syl 14 | . . . 4 |
20 | axcnex 7791 | . . . . . . 7 | |
21 | axresscn 7792 | . . . . . . 7 | |
22 | 20, 21 | ssexi 4114 | . . . . . 6 |
23 | 22 | inex2 4111 | . . . . 5 |
24 | eleq2 2228 | . . . . . . . 8 | |
25 | eleq2 2228 | . . . . . . . . 9 | |
26 | 25 | raleqbi1dv 2667 | . . . . . . . 8 |
27 | 24, 26 | anbi12d 465 | . . . . . . 7 |
28 | 27 | elabg 2867 | . . . . . 6 |
29 | nntopi.n | . . . . . . 7 | |
30 | intss1 3833 | . . . . . . 7 | |
31 | 29, 30 | eqsstrid 3183 | . . . . . 6 |
32 | 28, 31 | syl6bir 163 | . . . . 5 |
33 | 23, 32 | ax-mp 5 | . . . 4 |
34 | 7, 19, 33 | syl2an 287 | . . 3 |
35 | 34, 8 | sstrdi 3149 | . 2 |
36 | 3, 35 | sylan2br 286 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cab 2150 wral 2442 cvv 2721 cin 3110 wss 3111 cint 3818 (class class class)co 5836 cc 7742 cr 7743 c1 7745 caddc 7747 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-i1p 7399 df-iplp 7400 df-enr 7658 df-nr 7659 df-plr 7660 df-0r 7663 df-1r 7664 df-c 7750 df-1 7752 df-r 7754 df-add 7755 |
This theorem is referenced by: nnindnn 7825 |
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