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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 6350 | . 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 2689 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
4 | suceq 4324 | . . . . . . . . 9 ⊢ (𝑧 = 𝑤 → suc 𝑧 = suc 𝑤) | |
5 | eqid 2139 | . . . . . . . . 9 ⊢ (𝑧 ∈ V ↦ suc 𝑧) = (𝑧 ∈ V ↦ suc 𝑧) | |
6 | 3 | sucex 4415 | . . . . . . . . 9 ⊢ suc 𝑤 ∈ V |
7 | 4, 5, 6 | fvmpt 5498 | . . . . . . . 8 ⊢ (𝑤 ∈ V → ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) = suc 𝑤) |
8 | 3, 7 | ax-mp 5 | . . . . . . 7 ⊢ ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) = suc 𝑤 |
9 | 8 | eleq1i 2205 | . . . . . 6 ⊢ (((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ suc 𝑤 ∈ On) |
10 | 9 | ralbii 2441 | . . . . 5 ⊢ (∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ ∀𝑤 ∈ On suc 𝑤 ∈ On) |
11 | suceloni 4417 | . . . . 5 ⊢ (𝑤 ∈ On → suc 𝑤 ∈ On) | |
12 | 10, 11 | mprgbir 2490 | . . . 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 6283 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑧 ∈ V ↦ suc 𝑧), 𝐴)‘𝐵) ∈ On) |
15 | 1, 14 | eqeltrd 2216 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 Vcvv 2686 ↦ cmpt 3989 Oncon0 4285 suc csuc 4287 ‘cfv 5123 (class class class)co 5774 reccrdg 6266 +o coa 6310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-irdg 6267 df-oadd 6317 |
This theorem is referenced by: omcl 6357 omv2 6361 |
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