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Mirrors > Home > ILE Home > Th. List > 8nn | GIF version |
Description: 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
8nn | ⊢ 8 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-8 8943 | . 2 ⊢ 8 = (7 + 1) | |
2 | 7nn 9044 | . . 3 ⊢ 7 ∈ ℕ | |
3 | peano2nn 8890 | . . 3 ⊢ (7 ∈ ℕ → (7 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (7 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2243 | 1 ⊢ 8 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 (class class class)co 5853 1c1 7775 + caddc 7777 ℕcn 8878 7c7 8934 8c8 8935 |
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 df-5 8940 df-6 8941 df-7 8942 df-8 8943 |
This theorem is referenced by: 9nn 9046 8nn0 9158 ipndx 12556 ipid 12557 ipslid 12558 ipsstrd 12559 lgsval 13699 lgsfvalg 13700 lgsfcl2 13701 lgsval2lem 13705 lgsdir2lem1 13723 lgsdir2lem2 13724 lgsdir2lem3 13725 lgsdir2lem4 13726 lgsdir2lem5 13727 lgsdir2 13728 lgsne0 13733 |
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