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Mirrors > Home > ILE Home > Th. List > peano2nnnn | GIF version |
Description: A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8904 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7874). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
peano1nnnn.n | ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
Ref | Expression |
---|---|
peano2nnnn | ⊢ (𝐴 ∈ 𝑁 → (𝐴 + 1) ∈ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1nnnn.n | . . . . . 6 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
2 | 1 | eleq2i 2242 | . . . . 5 ⊢ (𝐴 ∈ 𝑁 ↔ 𝐴 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) |
3 | elintg 3848 | . . . . 5 ⊢ (𝐴 ∈ 𝑁 → (𝐴 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧)) | |
4 | 2, 3 | bitrid 192 | . . . 4 ⊢ (𝐴 ∈ 𝑁 → (𝐴 ∈ 𝑁 ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧)) |
5 | 4 | ibi 176 | . . 3 ⊢ (𝐴 ∈ 𝑁 → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧) |
6 | vex 2738 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | eleq2 2239 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧)) | |
8 | eleq2 2239 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧)) | |
9 | 8 | raleqbi1dv 2678 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
10 | 7, 9 | anbi12d 473 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))) |
11 | 6, 10 | elab 2879 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
12 | 11 | simprbi 275 | . . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) |
13 | oveq1 5872 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 + 1) = (𝐴 + 1)) | |
14 | 13 | eleq1d 2244 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑦 + 1) ∈ 𝑧 ↔ (𝐴 + 1) ∈ 𝑧)) |
15 | 14 | rspcva 2837 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → (𝐴 + 1) ∈ 𝑧) |
16 | 12, 15 | sylan2 286 | . . . . 5 ⊢ ((𝐴 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) → (𝐴 + 1) ∈ 𝑧) |
17 | 16 | expcom 116 | . . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝐴 ∈ 𝑧 → (𝐴 + 1) ∈ 𝑧)) |
18 | 17 | ralimia 2536 | . . 3 ⊢ (∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧 → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧) |
19 | 5, 18 | syl 14 | . 2 ⊢ (𝐴 ∈ 𝑁 → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧) |
20 | df-1 7794 | . . . . 5 ⊢ 1 = 〈1R, 0R〉 | |
21 | 1sr 7725 | . . . . . 6 ⊢ 1R ∈ R | |
22 | 0r 7724 | . . . . . 6 ⊢ 0R ∈ R | |
23 | opexg 4222 | . . . . . 6 ⊢ ((1R ∈ R ∧ 0R ∈ R) → 〈1R, 0R〉 ∈ V) | |
24 | 21, 22, 23 | mp2an 426 | . . . . 5 ⊢ 〈1R, 0R〉 ∈ V |
25 | 20, 24 | eqeltri 2248 | . . . 4 ⊢ 1 ∈ V |
26 | addvalex 7818 | . . . 4 ⊢ ((𝐴 ∈ 𝑁 ∧ 1 ∈ V) → (𝐴 + 1) ∈ V) | |
27 | 25, 26 | mpan2 425 | . . 3 ⊢ (𝐴 ∈ 𝑁 → (𝐴 + 1) ∈ V) |
28 | 1 | eleq2i 2242 | . . . 4 ⊢ ((𝐴 + 1) ∈ 𝑁 ↔ (𝐴 + 1) ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) |
29 | elintg 3848 | . . . 4 ⊢ ((𝐴 + 1) ∈ V → ((𝐴 + 1) ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧)) | |
30 | 28, 29 | bitrid 192 | . . 3 ⊢ ((𝐴 + 1) ∈ V → ((𝐴 + 1) ∈ 𝑁 ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧)) |
31 | 27, 30 | syl 14 | . 2 ⊢ (𝐴 ∈ 𝑁 → ((𝐴 + 1) ∈ 𝑁 ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧)) |
32 | 19, 31 | mpbird 167 | 1 ⊢ (𝐴 ∈ 𝑁 → (𝐴 + 1) ∈ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2146 {cab 2161 ∀wral 2453 Vcvv 2735 〈cop 3592 ∩ cint 3840 (class class class)co 5865 Rcnr 7271 0Rc0r 7272 1Rc1r 7273 1c1 7787 + caddc 7789 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-eprel 4283 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-1o 6407 df-2o 6408 df-oadd 6411 df-omul 6412 df-er 6525 df-ec 6527 df-qs 6531 df-ni 7278 df-pli 7279 df-mi 7280 df-lti 7281 df-plpq 7318 df-mpq 7319 df-enq 7321 df-nqqs 7322 df-plqqs 7323 df-mqqs 7324 df-1nqqs 7325 df-rq 7326 df-ltnqqs 7327 df-enq0 7398 df-nq0 7399 df-0nq0 7400 df-plq0 7401 df-mq0 7402 df-inp 7440 df-i1p 7441 df-iplp 7442 df-enr 7700 df-nr 7701 df-0r 7705 df-1r 7706 df-c 7792 df-1 7794 df-add 7797 |
This theorem is referenced by: nnindnn 7867 |
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