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Mirrors > Home > ILE Home > Th. List > oacl | GIF version |
Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.) |
Ref | Expression |
---|---|
oacl | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oav 6034 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (rec((𝑧 ∈ V ↦ suc 𝑧), 𝐴)‘𝐵)) | |
2 | oafnex 6024 | . . . 4 ⊢ (𝑧 ∈ V ↦ suc 𝑧) Fn V | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝐴 ∈ On → (𝑧 ∈ V ↦ suc 𝑧) Fn V) |
4 | id 19 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ On) | |
5 | vex 2560 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
6 | suceq 4139 | . . . . . . . . 9 ⊢ (𝑧 = 𝑤 → suc 𝑧 = suc 𝑤) | |
7 | eqid 2040 | . . . . . . . . 9 ⊢ (𝑧 ∈ V ↦ suc 𝑧) = (𝑧 ∈ V ↦ suc 𝑧) | |
8 | 5 | sucex 4225 | . . . . . . . . 9 ⊢ suc 𝑤 ∈ V |
9 | 6, 7, 8 | fvmpt 5249 | . . . . . . . 8 ⊢ (𝑤 ∈ V → ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) = suc 𝑤) |
10 | 5, 9 | ax-mp 7 | . . . . . . 7 ⊢ ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) = suc 𝑤 |
11 | 10 | eleq1i 2103 | . . . . . 6 ⊢ (((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ suc 𝑤 ∈ On) |
12 | 11 | ralbii 2330 | . . . . 5 ⊢ (∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ ∀𝑤 ∈ On suc 𝑤 ∈ On) |
13 | suceloni 4227 | . . . . 5 ⊢ (𝑤 ∈ On → suc 𝑤 ∈ On) | |
14 | 12, 13 | mprgbir 2379 | . . . 4 ⊢ ∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On |
15 | 14 | a1i 9 | . . 3 ⊢ (𝐴 ∈ On → ∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On) |
16 | 3, 4, 15 | rdgon 5973 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑧 ∈ V ↦ suc 𝑧), 𝐴)‘𝐵) ∈ On) |
17 | 1, 16 | eqeltrd 2114 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∀wral 2306 Vcvv 2557 ↦ cmpt 3818 Oncon0 4100 suc csuc 4102 Fn wfn 4897 ‘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-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: omcl 6041 omv2 6045 |
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