Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9345 | . 2 ⊢ (9 + 1) = ;10 | |
2 | 9nn 9046 | . . 3 ⊢ 9 ∈ ℕ | |
3 | peano2nn 8890 | . . 3 ⊢ (9 ∈ ℕ → (9 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (9 + 1) ∈ ℕ |
5 | 1, 4 | eqeltrri 2244 | 1 ⊢ ;10 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 (class class class)co 5853 0cc0 7774 1c1 7775 + caddc 7777 ℕcn 8878 9c9 8936 ;cdc 9343 |
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-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-1rid 7881 ax-0id 7882 ax-cnre 7885 |
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-rab 2457 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 df-9 8944 df-dec 9344 |
This theorem is referenced by: 10pos 9359 10re 9361 decnncl2 9366 declt 9370 decltc 9371 declti 9380 dec10p 9385 plendx 12573 pleid 12574 pleslid 12575 |
Copyright terms: Public domain | W3C validator |