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Theorem oecl 8573
Description: Closure law for ordinal exponentiation. Remark 2.8 of [Schloeder] p. 5. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
oecl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)

Proof of Theorem oecl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7438 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2 oe0m0 8556 . . . . . . . . 9 (∅ ↑o ∅) = 1o
3 1on 8516 . . . . . . . . 9 1o ∈ On
42, 3eqeltri 2834 . . . . . . . 8 (∅ ↑o ∅) ∈ On
51, 4eqeltrdi 2846 . . . . . . 7 (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
65adantl 481 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
7 oe0m1 8557 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
87biimpa 476 . . . . . . . 8 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
9 0elon 6439 . . . . . . . 8 ∅ ∈ On
108, 9eqeltrdi 2846 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
1110adantll 714 . . . . . 6 (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
126, 11oe0lem 8549 . . . . 5 ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
1312anidms 566 . . . 4 (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
14 oveq1 7437 . . . . 5 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
1514eleq1d 2823 . . . 4 (𝐴 = ∅ → ((𝐴o 𝐵) ∈ On ↔ (∅ ↑o 𝐵) ∈ On))
1613, 15imbitrrid 246 . . 3 (𝐴 = ∅ → (𝐵 ∈ On → (𝐴o 𝐵) ∈ On))
1716impcom 407 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴o 𝐵) ∈ On)
18 oveq2 7438 . . . . . . 7 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
1918eleq1d 2823 . . . . . 6 (𝑥 = ∅ → ((𝐴o 𝑥) ∈ On ↔ (𝐴o ∅) ∈ On))
20 oveq2 7438 . . . . . . 7 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
2120eleq1d 2823 . . . . . 6 (𝑥 = 𝑦 → ((𝐴o 𝑥) ∈ On ↔ (𝐴o 𝑦) ∈ On))
22 oveq2 7438 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
2322eleq1d 2823 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ∈ On ↔ (𝐴o suc 𝑦) ∈ On))
24 oveq2 7438 . . . . . . 7 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
2524eleq1d 2823 . . . . . 6 (𝑥 = 𝐵 → ((𝐴o 𝑥) ∈ On ↔ (𝐴o 𝐵) ∈ On))
26 oe0 8558 . . . . . . . 8 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2726, 3eqeltrdi 2846 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) ∈ On)
2827adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o ∅) ∈ On)
29 omcl 8572 . . . . . . . . . . 11 (((𝐴o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴o 𝑦) ·o 𝐴) ∈ On)
3029expcom 413 . . . . . . . . . 10 (𝐴 ∈ On → ((𝐴o 𝑦) ∈ On → ((𝐴o 𝑦) ·o 𝐴) ∈ On))
3130adantr 480 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝑦) ∈ On → ((𝐴o 𝑦) ·o 𝐴) ∈ On))
32 oesuc 8563 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
3332eleq1d 2823 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o suc 𝑦) ∈ On ↔ ((𝐴o 𝑦) ·o 𝐴) ∈ On))
3431, 33sylibrd 259 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝑦) ∈ On → (𝐴o suc 𝑦) ∈ On))
3534expcom 413 . . . . . . 7 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴o 𝑦) ∈ On → (𝐴o suc 𝑦) ∈ On)))
3635adantrd 491 . . . . . 6 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴o 𝑦) ∈ On → (𝐴o suc 𝑦) ∈ On)))
37 vex 3481 . . . . . . . . 9 𝑥 ∈ V
38 iunon 8377 . . . . . . . . 9 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴o 𝑦) ∈ On) → 𝑦𝑥 (𝐴o 𝑦) ∈ On)
3937, 38mpan 690 . . . . . . . 8 (∀𝑦𝑥 (𝐴o 𝑦) ∈ On → 𝑦𝑥 (𝐴o 𝑦) ∈ On)
40 oelim 8570 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4137, 40mpanlr1 706 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4241anasss 466 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4342an12s 649 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4443eleq1d 2823 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴o 𝑥) ∈ On ↔ 𝑦𝑥 (𝐴o 𝑦) ∈ On))
4539, 44imbitrrid 246 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦𝑥 (𝐴o 𝑦) ∈ On → (𝐴o 𝑥) ∈ On))
4645ex 412 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 (𝐴o 𝑦) ∈ On → (𝐴o 𝑥) ∈ On)))
4719, 21, 23, 25, 28, 36, 46tfinds3 7885 . . . . 5 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) ∈ On))
4847expd 415 . . . 4 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (𝐴o 𝐵) ∈ On)))
4948com12 32 . . 3 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝐴o 𝐵) ∈ On)))
5049imp31 417 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) ∈ On)
5117, 50oe0lem 8549 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  c0 4338   ciun 4995  Oncon0 6385  Lim wlim 6386  suc csuc 6387  (class class class)co 7430  1oc1o 8497   ·o comu 8502  o coe 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-omul 8509  df-oexp 8510
This theorem is referenced by:  oen0  8622  oeordi  8623  oeord  8624  oecan  8625  oeword  8626  oewordri  8628  oeworde  8629  oeordsuc  8630  oeoalem  8632  oeoa  8633  oeoelem  8634  oeoe  8635  oelimcl  8636  oeeulem  8637  oeeui  8638  oaabs2  8685  omabs  8687  cantnfle  9708  cantnflt  9709  cantnfp1  9718  cantnflem1d  9725  cantnflem1  9726  cantnflem2  9727  cantnflem3  9728  cantnflem4  9729  cantnf  9730  oemapwe  9731  cantnffval2  9732  cnfcomlem  9736  cnfcom  9737  cnfcom3lem  9740  cnfcom3  9741  infxpenc  10055  onexoegt  43232  oaomoecl  43267  oenassex  43307  cantnftermord  43309  cantnfresb  43313  oacl2g  43319  omabs2  43321  omcl2  43322  ofoaf  43344  ofoafo  43345
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