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Mirrors > Home > ILE Home > Th. List > oa0 | GIF version |
Description: Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oa0 | ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 4272 | . . 3 ⊢ ∅ ∈ On | |
2 | oav 6301 | . . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 +o ∅) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅)) | |
3 | 1, 2 | mpan2 419 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅)) |
4 | rdg0g 6236 | . 2 ⊢ (𝐴 ∈ On → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅) = 𝐴) | |
5 | 3, 4 | eqtrd 2145 | 1 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 ∈ wcel 1461 Vcvv 2655 ∅c0 3327 ↦ cmpt 3947 Oncon0 4243 suc csuc 4245 ‘cfv 5079 (class class class)co 5726 reccrdg 6217 +o coa 6261 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-iord 4246 df-on 4248 df-suc 4251 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-recs 6153 df-irdg 6218 df-oadd 6268 |
This theorem is referenced by: oa1suc 6314 oaword1 6318 nna0 6321 nna0r 6325 nnm0r 6326 |
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