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Theorem oecl 7662
Description: Closure law for ordinal exponentiation. (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 6698 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2 oe0m0 7645 . . . . . . . . 9 (∅ ↑𝑜 ∅) = 1𝑜
3 1on 7612 . . . . . . . . 9 1𝑜 ∈ On
42, 3eqeltri 2726 . . . . . . . 8 (∅ ↑𝑜 ∅) ∈ On
51, 4syl6eqel 2738 . . . . . . 7 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) ∈ On)
65adantl 481 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑𝑜 𝐵) ∈ On)
7 oe0m1 7646 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
87biimpa 500 . . . . . . . 8 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
9 0elon 5816 . . . . . . . 8 ∅ ∈ On
108, 9syl6eqel 2738 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
1110adantll 750 . . . . . 6 (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
126, 11oe0lem 7638 . . . . 5 ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
1312anidms 678 . . . 4 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
14 oveq1 6697 . . . . 5 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
1514eleq1d 2715 . . . 4 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ∈ On ↔ (∅ ↑𝑜 𝐵) ∈ On))
1613, 15syl5ibr 236 . . 3 (𝐴 = ∅ → (𝐵 ∈ On → (𝐴𝑜 𝐵) ∈ On))
1716impcom 445 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) ∈ On)
18 oveq2 6698 . . . . . . 7 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
1918eleq1d 2715 . . . . . 6 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 ∅) ∈ On))
20 oveq2 6698 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
2120eleq1d 2715 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 𝑦) ∈ On))
22 oveq2 6698 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
2322eleq1d 2715 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 suc 𝑦) ∈ On))
24 oveq2 6698 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
2524eleq1d 2715 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 𝐵) ∈ On))
26 oe0 7647 . . . . . . . 8 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
2726, 3syl6eqel 2738 . . . . . . 7 (𝐴 ∈ On → (𝐴𝑜 ∅) ∈ On)
2827adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴𝑜 ∅) ∈ On)
29 omcl 7661 . . . . . . . . . . 11 (((𝐴𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ On)
3029expcom 450 . . . . . . . . . 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 7652 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
3332eleq1d 2715 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 suc 𝑦) ∈ On ↔ ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ On))
3431, 33sylibrd 249 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 suc 𝑦) ∈ On))
3534expcom 450 . . . . . . 7 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 suc 𝑦) ∈ On)))
3635adantrd 483 . . . . . 6 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 suc 𝑦) ∈ On)))
37 vex 3234 . . . . . . . . 9 𝑥 ∈ V
38 iunon 7481 . . . . . . . . 9 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On) → 𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
3937, 38mpan 706 . . . . . . . 8 (∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On → 𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
40 oelim 7659 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4137, 40mpanlr1 722 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4241anasss 680 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4342an12s 860 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4443eleq1d 2715 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴𝑜 𝑥) ∈ On ↔ 𝑦𝑥 (𝐴𝑜 𝑦) ∈ On))
4539, 44syl5ibr 236 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 𝑥) ∈ On))
4645ex 449 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 𝑥) ∈ On)))
4719, 21, 23, 25, 28, 36, 46tfinds3 7106 . . . . 5 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) ∈ On))
4847expd 451 . . . 4 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (𝐴𝑜 𝐵) ∈ On)))
4948com12 32 . . 3 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝐴𝑜 𝐵) ∈ On)))
5049imp31 447 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) ∈ On)
5117, 50oe0lem 7638 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  c0 3948   ciun 4552  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690  1𝑜c1o 7598   ·𝑜 comu 7603  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by:  oen0  7711  oeordi  7712  oeord  7713  oecan  7714  oeword  7715  oewordri  7717  oeworde  7718  oeordsuc  7719  oeoalem  7721  oeoa  7722  oeoelem  7723  oeoe  7724  oelimcl  7725  oeeulem  7726  oeeui  7727  oaabs2  7770  omabs  7772  cantnfle  8606  cantnflt  8607  cantnfp1  8616  cantnflem1d  8623  cantnflem1  8624  cantnflem2  8625  cantnflem3  8626  cantnflem4  8627  cantnf  8628  oemapwe  8629  cantnffval2  8630  cnfcomlem  8634  cnfcom  8635  cnfcom3lem  8638  cnfcom3  8639  infxpenc  8879
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