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Mirrors > Home > ILE Home > Th. List > 1nn | GIF version |
Description: Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) |
Ref | Expression |
---|---|
1nn | ⊢ 1 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfnn2 8722 | . . . 4 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
2 | 1 | eleq2i 2206 | . . 3 ⊢ (1 ∈ ℕ ↔ 1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) |
3 | 1re 7765 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | elintg 3779 | . . . 4 ⊢ (1 ∈ ℝ → (1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧) |
6 | 2, 5 | bitri 183 | . 2 ⊢ (1 ∈ ℕ ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧) |
7 | vex 2689 | . . . 4 ⊢ 𝑧 ∈ V | |
8 | eleq2 2203 | . . . . 5 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧)) | |
9 | eleq2 2203 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧)) | |
10 | 9 | raleqbi1dv 2634 | . . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
11 | 8, 10 | anbi12d 464 | . . . 4 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))) |
12 | 7, 11 | elab 2828 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
13 | 12 | simplbi 272 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → 1 ∈ 𝑧) |
14 | 6, 13 | mprgbir 2490 | 1 ⊢ 1 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1480 {cab 2125 ∀wral 2416 ∩ cint 3771 (class class class)co 5774 ℝcr 7619 1c1 7621 + caddc 7623 ℕcn 8720 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-1re 7714 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-int 3772 df-inn 8721 |
This theorem is referenced by: nnind 8736 nn1suc 8739 2nn 8881 1nn0 8993 nn0p1nn 9016 1z 9080 neg1z 9086 elz2 9122 nneoor 9153 9p1e10 9184 indstr 9388 elnn1uz2 9401 zq 9418 qreccl 9434 fz01or 9891 exp3vallem 10294 exp1 10299 nnexpcl 10306 expnbnd 10415 3dec 10461 fac1 10475 faccl 10481 faclbnd3 10489 resqrexlemf1 10780 resqrexlemcalc3 10788 resqrexlemnmsq 10789 resqrexlemnm 10790 resqrexlemcvg 10791 resqrexlemglsq 10794 resqrexlemga 10795 sumsnf 11178 cvgratnnlemnexp 11293 cvgratnnlemfm 11298 cvgratnnlemrate 11299 cvgratnn 11300 eftlub 11396 eirraplem 11483 n2dvds1 11609 ndvdsp1 11629 gcd1 11675 bezoutr1 11721 ncoprmgcdne1b 11770 1nprm 11795 1idssfct 11796 isprm2lem 11797 qden1elz 11883 phicl2 11890 phi1 11895 phiprm 11899 exmidunben 11939 base0 12008 baseval 12011 baseid 12012 basendx 12013 basendxnn 12014 1strstrg 12057 2strstrg 12059 basendxnplusgndx 12065 basendxnmulrndx 12073 rngstrg 12074 lmodstrd 12092 topgrpstrd 12110 setsmsdsg 12649 trilpolemgt1 13232 |
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