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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 8981 | . 2 ⊢ 3 = (2 + 1) | |
2 | 2nn 9082 | . . 3 ⊢ 2 ∈ ℕ | |
3 | peano2nn 8933 | . . 3 ⊢ (2 ∈ ℕ → (2 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (2 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2250 | 1 ⊢ 3 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 (class class class)co 5877 1c1 7814 + caddc 7816 ℕcn 8921 2c2 8972 3c3 8973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4123 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-iota 5180 df-fv 5226 df-ov 5880 df-inn 8922 df-2 8980 df-3 8981 |
This theorem is referenced by: 4nn 9084 3nn0 9196 3z 9284 ige3m2fz 10051 sin01bnd 11767 3lcm2e6woprm 12088 3lcm2e6 12162 mulrndx 12590 mulridx 12591 mulrslid 12592 rngstrg 12595 unifndx 12682 unifid 12683 unifndxnn 12684 slotsdifunifndx 12688 cnfldstr 13542 tangtx 14344 lgsdir2lem1 14514 lgsdir2lem5 14518 |
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