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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 8580 | . 2 ⊢ 3 = (2 + 1) | |
2 | 2nn 8675 | . . 3 ⊢ 2 ∈ ℕ | |
3 | peano2nn 8532 | . . 3 ⊢ (2 ∈ ℕ → (2 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ (2 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2167 | 1 ⊢ 3 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1445 (class class class)co 5690 1c1 7448 + caddc 7450 ℕcn 8520 2c2 8571 3c3 8572 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-cnex 7533 ax-resscn 7534 ax-1re 7536 ax-addrcl 7539 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-iota 5014 df-fv 5057 df-ov 5693 df-inn 8521 df-2 8579 df-3 8580 |
This theorem is referenced by: 4nn 8677 3nn0 8789 3z 8877 ige3m2fz 9612 sin01bnd 11197 3lcm2e6woprm 11495 3lcm2e6 11566 mulrndx 11753 mulrid 11754 mulrslid 11755 rngstrg 11758 |
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