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Theorem oacl 6261
 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.)
Assertion
Ref Expression
oacl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)

Proof of Theorem oacl
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef 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)
63sucex 4344 . . . . . . . . 9 suc 𝑤 ∈ V
74, 5, 6fvmpt 5416 . . . . . . . 8 (𝑤 ∈ V → ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) = suc 𝑤)
83, 7ax-mp 7 . . . . . . 7 ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) = suc 𝑤
98eleq1i 2160 . . . . . 6 (((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ suc 𝑤 ∈ On)
109ralbii 2395 . . . . 5 (∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On ↔ ∀𝑤 ∈ On suc 𝑤 ∈ On)
11 suceloni 4346 . . . . 5 (𝑤 ∈ On → suc 𝑤 ∈ On)
1210, 11mprgbir 2444 . . . 4 𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On
1312a1i 9 . . 3 (𝐴 ∈ On → ∀𝑤 ∈ On ((𝑧 ∈ V ↦ suc 𝑧)‘𝑤) ∈ On)
142, 13rdgon 6189 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑧 ∈ V ↦ suc 𝑧), 𝐴)‘𝐵) ∈ On)
151, 14eqeltrd 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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