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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 6255 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑧 ∈ V ↦ suc 𝑧), 𝐴)‘𝐵)) | |
2 | id 19 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ On) | |
3 | vex 2636 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
4 | suceq 4253 | . . . . . . . . 9 ⊢ (𝑧 = 𝑤 → suc 𝑧 = suc 𝑤) | |
5 | eqid 2095 | . . . . . . . . 9 ⊢ (𝑧 ∈ V ↦ suc 𝑧) = (𝑧 ∈ V ↦ suc 𝑧) | |
6 | 3 | sucex 4344 | . . . . . . . . 9 ⊢ suc 𝑤 ∈ V |
7 | 4, 5, 6 | fvmpt 5416 | . . . . . . . 8 ⊢ (𝑤 ∈ V → ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) = suc 𝑤) |
8 | 3, 7 | ax-mp 7 | . . . . . . 7 ⊢ ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) = suc 𝑤 |
9 | 8 | eleq1i 2160 | . . . . . 6 ⊢ (((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ suc 𝑤 ∈ On) |
10 | 9 | ralbii 2395 | . . . . 5 ⊢ (∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ ∀𝑤 ∈ On suc 𝑤 ∈ On) |
11 | suceloni 4346 | . . . . 5 ⊢ (𝑤 ∈ On → suc 𝑤 ∈ On) | |
12 | 10, 11 | mprgbir 2444 | . . . 4 ⊢ ∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On |
13 | 12 | a1i 9 | . . 3 ⊢ (𝐴 ∈ On → ∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On) |
14 | 2, 13 | rdgon 6189 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑧 ∈ V ↦ suc 𝑧), 𝐴)‘𝐵) ∈ On) |
15 | 1, 14 | eqeltrd 2171 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 ∀wral 2370 Vcvv 2633 ↦ cmpt 3921 Oncon0 4214 suc csuc 4216 ‘cfv 5049 (class class class)co 5690 reccrdg 6172 +o coa 6216 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-suc 4222 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-recs 6108 df-irdg 6173 df-oadd 6223 |
This theorem is referenced by: omcl 6262 omv2 6266 |
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