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Mirrors > Home > ILE Home > Th. List > 9p2e11 | GIF version |
Description: 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9p2e11 | ⊢ (9 + 2) = ;11 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 7981 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 1nn0 7973 | . 2 ⊢ 1 ∈ ℕ0 | |
3 | 0nn0 7972 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | df-2 7753 | . 2 ⊢ 2 = (1 + 1) | |
5 | 1e0p1 8171 | . 2 ⊢ 1 = (0 + 1) | |
6 | df-10 7761 | . . 3 ⊢ 10 = (9 + 1) | |
7 | dec10 8173 | . . 3 ⊢ 10 = ;10 | |
8 | 6, 7 | eqtr3i 2059 | . 2 ⊢ (9 + 1) = ;10 |
9 | 1, 2, 3, 4, 5, 8 | 6p5lem 8192 | 1 ⊢ (9 + 2) = ;11 |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 (class class class)co 5455 0cc0 6711 1c1 6712 + caddc 6714 2c2 7744 9c9 7751 10c10 7752 ;cdc 8144 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-sub 6981 df-inn 7696 df-2 7753 df-3 7754 df-4 7755 df-5 7756 df-6 7757 df-7 7758 df-8 7759 df-9 7760 df-10 7761 df-n0 7958 df-dec 8145 |
This theorem is referenced by: 9p3e12 8206 9t9e81 8245 |
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