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Mirrors > Home > ILE Home > Th. List > 9p3e12 | GIF version |
Description: 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9p3e12 | ⊢ (9 + 3) = ;12 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 9008 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 2nn0 9001 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | 1nn0 9000 | . 2 ⊢ 1 ∈ ℕ0 | |
4 | df-3 8787 | . 2 ⊢ 3 = (2 + 1) | |
5 | df-2 8786 | . 2 ⊢ 2 = (1 + 1) | |
6 | 9p2e11 9275 | . 2 ⊢ (9 + 2) = ;11 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 9258 | 1 ⊢ (9 + 3) = ;12 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 (class class class)co 5774 1c1 7628 + caddc 7630 2c2 8778 3c3 8779 9c9 8785 ;cdc 9189 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-addcom 7727 ax-mulcom 7728 ax-addass 7729 ax-mulass 7730 ax-distr 7731 ax-i2m1 7732 ax-1rid 7734 ax-0id 7735 ax-rnegex 7736 ax-cnre 7738 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7942 df-inn 8728 df-2 8786 df-3 8787 df-4 8788 df-5 8789 df-6 8790 df-7 8791 df-8 8792 df-9 8793 df-n0 8985 df-dec 9190 |
This theorem is referenced by: 9p4e13 9277 9t8e72 9316 |
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