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Mirrors > Home > ILE Home > Th. List > 4nn | GIF version |
Description: 4 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
Ref | Expression |
---|---|
4nn | ⊢ 4 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-4 8544 | . 2 ⊢ 4 = (3 + 1) | |
2 | 3nn 8639 | . . 3 ⊢ 3 ∈ ℕ | |
3 | peano2nn 8495 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ (3 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2161 | 1 ⊢ 4 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1439 (class class class)co 5666 1c1 7412 + caddc 7414 ℕcn 8483 3c3 8535 4c4 8536 |
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-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-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-cnex 7497 ax-resscn 7498 ax-1re 7500 ax-addrcl 7503 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-iota 4993 df-fv 5036 df-ov 5669 df-inn 8484 df-2 8542 df-3 8543 df-4 8544 |
This theorem is referenced by: 5nn 8641 4nn0 8753 4z 8841 fldiv4p1lem1div2 9773 iexpcyc 10120 resqrexlemnmsq 10511 ef01bndlem 11108 flodddiv4 11273 flodddiv4t2lthalf 11276 6lcm4e12 11408 starvndx 11674 starvid 11675 starvslid 11676 srngstrd 11677 |
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