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Mirrors > Home > ILE Home > Th. List > 9p4e13 | GIF version |
Description: 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9p4e13 | ⊢ (9 + 4) = ;13 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 8758 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 3nn0 8752 | . 2 ⊢ 3 ∈ ℕ0 | |
3 | 2nn0 8751 | . 2 ⊢ 2 ∈ ℕ0 | |
4 | df-4 8544 | . 2 ⊢ 4 = (3 + 1) | |
5 | df-3 8543 | . 2 ⊢ 3 = (2 + 1) | |
6 | 9p3e12 9025 | . 2 ⊢ (9 + 3) = ;12 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 9007 | 1 ⊢ (9 + 4) = ;13 |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 (class class class)co 5666 1c1 7412 + caddc 7414 2c2 8534 3c3 8535 4c4 8536 9c9 8541 ;cdc 8938 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-sub 7716 df-inn 8484 df-2 8542 df-3 8543 df-4 8544 df-5 8545 df-6 8546 df-7 8547 df-8 8548 df-9 8549 df-n0 8735 df-dec 8939 |
This theorem is referenced by: 9p5e14 9027 9t7e63 9064 |
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