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Mirrors > Home > ILE Home > Th. List > peano2nnnn | Unicode version |
Description: A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8845 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7820). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
peano1nnnn.n |
Ref | Expression |
---|---|
peano2nnnn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1nnnn.n | . . . . . 6 | |
2 | 1 | eleq2i 2224 | . . . . 5 |
3 | elintg 3815 | . . . . 5 | |
4 | 2, 3 | syl5bb 191 | . . . 4 |
5 | 4 | ibi 175 | . . 3 |
6 | vex 2715 | . . . . . . . 8 | |
7 | eleq2 2221 | . . . . . . . . 9 | |
8 | eleq2 2221 | . . . . . . . . . 10 | |
9 | 8 | raleqbi1dv 2660 | . . . . . . . . 9 |
10 | 7, 9 | anbi12d 465 | . . . . . . . 8 |
11 | 6, 10 | elab 2856 | . . . . . . 7 |
12 | 11 | simprbi 273 | . . . . . 6 |
13 | oveq1 5831 | . . . . . . . 8 | |
14 | 13 | eleq1d 2226 | . . . . . . 7 |
15 | 14 | rspcva 2814 | . . . . . 6 |
16 | 12, 15 | sylan2 284 | . . . . 5 |
17 | 16 | expcom 115 | . . . 4 |
18 | 17 | ralimia 2518 | . . 3 |
19 | 5, 18 | syl 14 | . 2 |
20 | df-1 7740 | . . . . 5 | |
21 | 1sr 7671 | . . . . . 6 | |
22 | 0r 7670 | . . . . . 6 | |
23 | opexg 4188 | . . . . . 6 | |
24 | 21, 22, 23 | mp2an 423 | . . . . 5 |
25 | 20, 24 | eqeltri 2230 | . . . 4 |
26 | addvalex 7764 | . . . 4 | |
27 | 25, 26 | mpan2 422 | . . 3 |
28 | 1 | eleq2i 2224 | . . . 4 |
29 | elintg 3815 | . . . 4 | |
30 | 28, 29 | syl5bb 191 | . . 3 |
31 | 27, 30 | syl 14 | . 2 |
32 | 19, 31 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 cab 2143 wral 2435 cvv 2712 cop 3563 cint 3807 (class class class)co 5824 cnr 7217 c0r 7218 c1r 7219 c1 7733 caddc 7735 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-1o 6363 df-2o 6364 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-lti 7227 df-plpq 7264 df-mpq 7265 df-enq 7267 df-nqqs 7268 df-plqqs 7269 df-mqqs 7270 df-1nqqs 7271 df-rq 7272 df-ltnqqs 7273 df-enq0 7344 df-nq0 7345 df-0nq0 7346 df-plq0 7347 df-mq0 7348 df-inp 7386 df-i1p 7387 df-iplp 7388 df-enr 7646 df-nr 7647 df-0r 7651 df-1r 7652 df-c 7738 df-1 7740 df-add 7743 |
This theorem is referenced by: nnindnn 7813 |
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