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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 8939 | . 2 ⊢ 4 = (3 + 1) | |
2 | 3nn 9040 | . . 3 ⊢ 3 ∈ ℕ | |
3 | peano2nn 8890 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (3 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2243 | 1 ⊢ 4 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 (class class class)co 5853 1c1 7775 + caddc 7777 ℕcn 8878 3c3 8930 4c4 8931 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4107 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 |
This theorem is referenced by: 5nn 9042 4nn0 9154 4z 9242 fldiv4p1lem1div2 10261 iexpcyc 10580 resqrexlemnmsq 10981 ef01bndlem 11719 flodddiv4 11893 flodddiv4t2lthalf 11896 6lcm4e12 12041 starvndx 12537 starvid 12538 starvslid 12539 srngstrd 12540 dveflem 13481 tan4thpi 13556 |
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