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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 9213 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 3nn0 9207 | . 2 ⊢ 3 ∈ ℕ0 | |
3 | 2nn0 9206 | . 2 ⊢ 2 ∈ ℕ0 | |
4 | df-4 8993 | . 2 ⊢ 4 = (3 + 1) | |
5 | df-3 8992 | . 2 ⊢ 3 = (2 + 1) | |
6 | 9p3e12 9484 | . 2 ⊢ (9 + 3) = ;12 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 9466 | 1 ⊢ (9 + 4) = ;13 |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 (class class class)co 5888 1c1 7825 + caddc 7827 2c2 8983 3c3 8984 4c4 8985 9c9 8990 ;cdc 9397 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8143 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-9 8998 df-n0 9190 df-dec 9398 |
This theorem is referenced by: 9p5e14 9486 9t7e63 9523 |
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