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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 8497 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7498). (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 2155 | . . . . 5 ⊢ (𝐴 ∈ 𝑁 ↔ 𝐴 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) |
3 | elintg 3704 | . . . . 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 | syl5bb 191 | . . . 4 ⊢ (𝐴 ∈ 𝑁 → (𝐴 ∈ 𝑁 ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧)) |
5 | 4 | ibi 175 | . . 3 ⊢ (𝐴 ∈ 𝑁 → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧) |
6 | vex 2625 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | eleq2 2152 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧)) | |
8 | eleq2 2152 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧)) | |
9 | 8 | raleqbi1dv 2573 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
10 | 7, 9 | anbi12d 458 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))) |
11 | 6, 10 | elab 2763 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
12 | 11 | simprbi 270 | . . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) |
13 | oveq1 5675 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 + 1) = (𝐴 + 1)) | |
14 | 13 | eleq1d 2157 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑦 + 1) ∈ 𝑧 ↔ (𝐴 + 1) ∈ 𝑧)) |
15 | 14 | rspcva 2723 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → (𝐴 + 1) ∈ 𝑧) |
16 | 12, 15 | sylan2 281 | . . . . 5 ⊢ ((𝐴 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) → (𝐴 + 1) ∈ 𝑧) |
17 | 16 | expcom 115 | . . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝐴 ∈ 𝑧 → (𝐴 + 1) ∈ 𝑧)) |
18 | 17 | ralimia 2437 | . . 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 7421 | . . . . 5 ⊢ 1 = 〈1R, 0R〉 | |
21 | 1sr 7360 | . . . . . 6 ⊢ 1R ∈ R | |
22 | 0r 7359 | . . . . . 6 ⊢ 0R ∈ R | |
23 | opexg 4066 | . . . . . 6 ⊢ ((1R ∈ R ∧ 0R ∈ R) → 〈1R, 0R〉 ∈ V) | |
24 | 21, 22, 23 | mp2an 418 | . . . . 5 ⊢ 〈1R, 0R〉 ∈ V |
25 | 20, 24 | eqeltri 2161 | . . . 4 ⊢ 1 ∈ V |
26 | addvalex 7444 | . . . 4 ⊢ ((𝐴 ∈ 𝑁 ∧ 1 ∈ V) → (𝐴 + 1) ∈ V) | |
27 | 25, 26 | mpan2 417 | . . 3 ⊢ (𝐴 ∈ 𝑁 → (𝐴 + 1) ∈ V) |
28 | 1 | eleq2i 2155 | . . . 4 ⊢ ((𝐴 + 1) ∈ 𝑁 ↔ (𝐴 + 1) ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) |
29 | elintg 3704 | . . . 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 | syl5bb 191 | . . 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 166 | 1 ⊢ (𝐴 ∈ 𝑁 → (𝐴 + 1) ∈ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∈ wcel 1439 {cab 2075 ∀wral 2360 Vcvv 2622 〈cop 3455 ∩ cint 3696 (class class class)co 5668 Rcnr 6919 0Rc0r 6920 1Rc1r 6921 1c1 7414 + caddc 7416 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-eprel 4127 df-id 4131 df-po 4134 df-iso 4135 df-iord 4204 df-on 4206 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-recs 6086 df-irdg 6151 df-1o 6197 df-2o 6198 df-oadd 6201 df-omul 6202 df-er 6308 df-ec 6310 df-qs 6314 df-ni 6926 df-pli 6927 df-mi 6928 df-lti 6929 df-plpq 6966 df-mpq 6967 df-enq 6969 df-nqqs 6970 df-plqqs 6971 df-mqqs 6972 df-1nqqs 6973 df-rq 6974 df-ltnqqs 6975 df-enq0 7046 df-nq0 7047 df-0nq0 7048 df-plq0 7049 df-mq0 7050 df-inp 7088 df-i1p 7089 df-iplp 7090 df-enr 7335 df-nr 7336 df-0r 7340 df-1r 7341 df-c 7419 df-1 7421 df-add 7424 |
This theorem is referenced by: nnindnn 7491 |
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