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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 6358 | . 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 2692 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
4 | suceq 4332 | . . . . . . . . 9 ⊢ (𝑧 = 𝑤 → suc 𝑧 = suc 𝑤) | |
5 | eqid 2140 | . . . . . . . . 9 ⊢ (𝑧 ∈ V ↦ suc 𝑧) = (𝑧 ∈ V ↦ suc 𝑧) | |
6 | 3 | sucex 4423 | . . . . . . . . 9 ⊢ suc 𝑤 ∈ V |
7 | 4, 5, 6 | fvmpt 5506 | . . . . . . . 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 2206 | . . . . . 6 ⊢ (((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ suc 𝑤 ∈ On) |
10 | 9 | ralbii 2444 | . . . . 5 ⊢ (∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ ∀𝑤 ∈ On suc 𝑤 ∈ On) |
11 | suceloni 4425 | . . . . 5 ⊢ (𝑤 ∈ On → suc 𝑤 ∈ On) | |
12 | 10, 11 | mprgbir 2493 | . . . 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 6291 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑧 ∈ V ↦ suc 𝑧), 𝐴)‘𝐵) ∈ On) |
15 | 1, 14 | eqeltrd 2217 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ∀wral 2417 Vcvv 2689 ↦ cmpt 3997 Oncon0 4293 suc csuc 4295 ‘cfv 5131 (class class class)co 5782 reccrdg 6274 +o coa 6318 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-recs 6210 df-irdg 6275 df-oadd 6325 |
This theorem is referenced by: omcl 6365 omv2 6369 |
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