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Theorem oacl 8485
Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. Remark 2.8 of [Schloeder] p. 5. (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 7369 . . . 4 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
21eleq1d 2819 . . 3 (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o ∅) ∈ On))
3 oveq2 7369 . . . 4 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
43eleq1d 2819 . . 3 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝑦) ∈ On))
5 oveq2 7369 . . . 4 (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
65eleq1d 2819 . . 3 (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o suc 𝑦) ∈ On))
7 oveq2 7369 . . . 4 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
87eleq1d 2819 . . 3 (𝑥 = 𝐵 → ((𝐴 +o 𝑥) ∈ On ↔ (𝐴 +o 𝐵) ∈ On))
9 oa0 8466 . . . . 5 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
109eleq1d 2819 . . . 4 (𝐴 ∈ On → ((𝐴 +o ∅) ∈ On ↔ 𝐴 ∈ On))
1110ibir 268 . . 3 (𝐴 ∈ On → (𝐴 +o ∅) ∈ On)
12 onsuc 7750 . . . . 5 ((𝐴 +o 𝑦) ∈ On → suc (𝐴 +o 𝑦) ∈ On)
13 oasuc 8474 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
1413eleq1d 2819 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o suc 𝑦) ∈ On ↔ suc (𝐴 +o 𝑦) ∈ On))
1512, 14imbitrrid 245 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On))
1615expcom 415 . . 3 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑦) ∈ On → (𝐴 +o suc 𝑦) ∈ On)))
17 vex 3451 . . . . . 6 𝑥 ∈ V
18 iunon 8289 . . . . . 6 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴 +o 𝑦) ∈ On) → 𝑦𝑥 (𝐴 +o 𝑦) ∈ On)
1917, 18mpan 689 . . . . 5 (∀𝑦𝑥 (𝐴 +o 𝑦) ∈ On → 𝑦𝑥 (𝐴 +o 𝑦) ∈ On)
20 oalim 8482 . . . . . . 7 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = 𝑦𝑥 (𝐴 +o 𝑦))
2117, 20mpanr1 702 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 +o 𝑥) = 𝑦𝑥 (𝐴 +o 𝑦))
2221eleq1d 2819 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝑥) ∈ On ↔ 𝑦𝑥 (𝐴 +o 𝑦) ∈ On))
2319, 22imbitrrid 245 . . . 4 ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On))
2423expcom 415 . . 3 (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦𝑥 (𝐴 +o 𝑦) ∈ On → (𝐴 +o 𝑥) ∈ On)))
252, 4, 6, 8, 11, 16, 24tfinds3 7805 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) ∈ On))
2625impcom 409 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3447  c0 4286   ciun 4958  Oncon0 6321  Lim wlim 6322  suc csuc 6323  (class class class)co 7361   +o coa 8413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-oadd 8420
This theorem is referenced by:  omcl  8486  oaord  8498  oacan  8499  oaword  8500  oawordri  8501  oawordeulem  8505  oalimcl  8511  oaass  8512  oaf1o  8514  odi  8530  omopth2  8535  oeoalem  8547  oeoa  8548  oancom  9595  cantnfvalf  9609  dfac12lem2  10088  djunum  10139  wunex3  10685  rdgeqoa  35891  oaomoecl  41660  oawordex2  41708  omabs2  41714  ofoafg  41717  oaun3lem1  41737  oaun3lem2  41738  oaun3lem3  41739  oaun3lem4  41740  oadif1  41743  oaun2  41744  oaun3  41745  naddgeoa  41758  naddwordnexlem3  41763  oawordex3  41764  naddwordnexlem4  41765  oa1cl  41811
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