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Mirrors > Home > ILE Home > Th. List > 10nn | GIF version |
Description: 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
10nn | ⊢ ;10 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9p1e10 9415 | . 2 ⊢ (9 + 1) = ;10 | |
2 | 9nn 9116 | . . 3 ⊢ 9 ∈ ℕ | |
3 | peano2nn 8960 | . . 3 ⊢ (9 ∈ ℕ → (9 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (9 + 1) ∈ ℕ |
5 | 1, 4 | eqeltrri 2263 | 1 ⊢ ;10 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 (class class class)co 5895 0cc0 7840 1c1 7841 + caddc 7843 ℕcn 8948 9c9 9006 ;cdc 9413 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4136 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-1rid 7947 ax-0id 7948 ax-cnre 7951 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5898 df-inn 8949 df-2 9007 df-3 9008 df-4 9009 df-5 9010 df-6 9011 df-7 9012 df-8 9013 df-9 9014 df-dec 9414 |
This theorem is referenced by: 10pos 9429 10re 9431 decnncl2 9436 declt 9440 decltc 9441 declti 9450 dec10p 9455 plendx 12708 pleid 12709 pleslid 12710 plendxnn 12711 |
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