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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 4129 | . . 3 ⊢ ∅ ∈ On | |
2 | oav 6034 | . . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 +𝑜 ∅) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅)) | |
3 | 1, 2 | mpan2 401 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅)) |
4 | rdg0g 5975 | . 2 ⊢ (𝐴 ∈ On → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅) = 𝐴) | |
5 | 3, 4 | eqtrd 2072 | 1 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∅c0 3224 ↦ cmpt 3818 Oncon0 4100 suc csuc 4102 ‘cfv 4902 (class class class)co 5512 reccrdg 5956 +𝑜 coa 5998 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-recs 5920 df-irdg 5957 df-oadd 6005 |
This theorem is referenced by: oa1suc 6047 oaword1 6050 nna0 6053 nna0r 6057 nnm0r 6058 |
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