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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 5973 | . 2 ⊢ ((A ∈ On ∧ B ∈ On) → (A +𝑜 B) = (rec((z ∈ V ↦ suc z), A)‘B)) | |
2 | oafnex 5963 | . . . 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 2554 | . . . . . . . 8 ⊢ w ∈ V | |
6 | suceq 4105 | . . . . . . . . 9 ⊢ (z = w → suc z = suc w) | |
7 | eqid 2037 | . . . . . . . . 9 ⊢ (z ∈ V ↦ suc z) = (z ∈ V ↦ suc z) | |
8 | 5 | sucex 4191 | . . . . . . . . 9 ⊢ suc w ∈ V |
9 | 6, 7, 8 | fvmpt 5192 | . . . . . . . 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 2100 | . . . . . 6 ⊢ (((z ∈ V ↦ suc z)‘w) ∈ On ↔ suc w ∈ On) |
12 | 11 | ralbii 2324 | . . . . 5 ⊢ (∀w ∈ On ((z ∈ V ↦ suc z)‘w) ∈ On ↔ ∀w ∈ On suc w ∈ On) |
13 | suceloni 4193 | . . . . 5 ⊢ (w ∈ On → suc w ∈ On) | |
14 | 12, 13 | mprgbir 2373 | . . . 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 5913 | . 2 ⊢ ((A ∈ On ∧ B ∈ On) → (rec((z ∈ V ↦ suc z), A)‘B) ∈ On) |
17 | 1, 16 | eqeltrd 2111 | 1 ⊢ ((A ∈ On ∧ B ∈ On) → (A +𝑜 B) ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∀wral 2300 Vcvv 2551 ↦ cmpt 3809 Oncon0 4066 suc csuc 4068 Fn wfn 4840 ‘cfv 4845 (class class class)co 5455 reccrdg 5896 +𝑜 coa 5937 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-recs 5861 df-irdg 5897 df-oadd 5944 |
This theorem is referenced by: omcl 5980 omv2 5984 |
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