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Theorem oacl 8138
Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
Assertion
Ref Expression
oacl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)

Proof of Theorem oacl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7141 . . . 4 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
21eleq1d 2895 . . 3 (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o ∅) ∈ On))
3 oveq2 7141 . . . 4 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
43eleq1d 2895 . . 3 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝑦) ∈ On))
5 oveq2 7141 . . . 4 (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
65eleq1d 2895 . . 3 (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o suc 𝑦) ∈ On))
7 oveq2 7141 . . . 4 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
87eleq1d 2895 . . 3 (𝑥 = 𝐵 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝐵) ∈ On))
9 oa0 8119 . . . . 5 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
109eleq1d 2895 . . . 4 (𝐴 ∈ On → ((𝐴 +o ∅) ∈ On ↔ 𝐴 ∈ On))
1110ibir 270 . . 3 (𝐴 ∈ On → (𝐴 +o ∅) ∈ On)
12 suceloni 7506 . . . . 5 ((𝐴 +o 𝑦) ∈ On → suc (𝐴 +o 𝑦) ∈ On)
13 oasuc 8127 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
1413eleq1d 2895 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o suc 𝑦) ∈ On ↔ suc (𝐴 +o 𝑦) ∈ On))
1512, 14syl5ibr 248 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On))
1615expcom 416 . . 3 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On)))
17 vex 3476 . . . . . 6 𝑥 ∈ V
18 iunon 7954 . . . . . 6 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴 +o 𝑦) ∈ On) → 𝑦𝑥 (𝐴 +o 𝑦) ∈ On)
1917, 18mpan 688 . . . . 5 (∀𝑦𝑥 (𝐴 +o 𝑦) ∈ On → 𝑦𝑥 (𝐴 +o 𝑦) ∈ On)
20 oalim 8135 . . . . . . 7 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = 𝑦𝑥 (𝐴 +o 𝑦))
2117, 20mpanr1 701 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 +o 𝑥) = 𝑦𝑥 (𝐴 +o 𝑦))
2221eleq1d 2895 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝑥) ∈ On ↔ 𝑦𝑥 (𝐴 +o 𝑦) ∈ On))
2319, 22syl5ibr 248 . . . 4 ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On))
2423expcom 416 . . 3 (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On)))
252, 4, 6, 8, 11, 16, 24tfinds3 7557 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) ∈ On))
2625impcom 410 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  wral 3125  Vcvv 3473  c0 4269   ciun 4895  Oncon0 6167  Lim wlim 6168  suc csuc 6169  (class class class)co 7133   +o coa 8077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-oadd 8084
This theorem is referenced by:  omcl  8139  oaord  8151  oacan  8152  oaword  8153  oawordri  8154  oawordeulem  8158  oalimcl  8164  oaass  8165  oaf1o  8167  odi  8183  omopth2  8188  oeoalem  8200  oeoa  8201  oancom  9092  cantnfvalf  9106  dfac12lem2  9548  djunum  9599  wunex3  10141  rdgeqoa  34668
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