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Mirrors > Home > ILE Home > Th. List > 7nn | GIF version |
Description: 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
7nn | ⊢ 7 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-7 8959 | . 2 ⊢ 7 = (6 + 1) | |
2 | 6nn 9060 | . . 3 ⊢ 6 ∈ ℕ | |
3 | peano2nn 8907 | . . 3 ⊢ (6 ∈ ℕ → (6 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (6 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2250 | 1 ⊢ 7 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 (class class class)co 5868 1c1 7790 + caddc 7792 ℕcn 8895 6c6 8950 7c7 8951 |
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 4118 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
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 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-iota 5173 df-fv 5219 df-ov 5871 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-5 8957 df-6 8958 df-7 8959 |
This theorem is referenced by: 8nn 9062 7nn0 9174 lgsval 14038 lgsfvalg 14039 lgsfcl2 14040 lgsval2lem 14044 lgsdir2lem1 14062 lgsdir2lem3 14064 lgsdir2 14067 lgsne0 14072 |
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