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Mirrors > Home > ILE Home > Th. List > 3nn | GIF version |
Description: 3 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
Ref | Expression |
---|---|
3nn | ⊢ 3 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3 8931 | . 2 ⊢ 3 = (2 + 1) | |
2 | 2nn 9032 | . . 3 ⊢ 2 ∈ ℕ | |
3 | peano2nn 8883 | . . 3 ⊢ (2 ∈ ℕ → (2 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (2 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2243 | 1 ⊢ 3 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 (class class class)co 5851 1c1 7768 + caddc 7770 ℕcn 8871 2c2 8922 3c3 8923 |
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 4105 ax-cnex 7858 ax-resscn 7859 ax-1re 7861 ax-addrcl 7864 |
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 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5854 df-inn 8872 df-2 8930 df-3 8931 |
This theorem is referenced by: 4nn 9034 3nn0 9146 3z 9234 ige3m2fz 9998 sin01bnd 11713 3lcm2e6woprm 12033 3lcm2e6 12107 mulrndx 12521 mulrid 12522 mulrslid 12523 rngstrg 12526 tangtx 13518 lgsdir2lem1 13688 lgsdir2lem5 13692 |
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