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Theorem oecl 7771
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 6801 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2 oe0m0 7754 . . . . . . . . 9 (∅ ↑𝑜 ∅) = 1𝑜
3 1on 7720 . . . . . . . . 9 1𝑜 ∈ On
42, 3eqeltri 2846 . . . . . . . 8 (∅ ↑𝑜 ∅) ∈ On
51, 4syl6eqel 2858 . . . . . . 7 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) ∈ On)
65adantl 467 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑𝑜 𝐵) ∈ On)
7 oe0m1 7755 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
87biimpa 462 . . . . . . . 8 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
9 0elon 5921 . . . . . . . 8 ∅ ∈ On
108, 9syl6eqel 2858 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
1110adantll 693 . . . . . 6 (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
126, 11oe0lem 7747 . . . . 5 ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
1312anidms 556 . . . 4 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
14 oveq1 6800 . . . . 5 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
1514eleq1d 2835 . . . 4 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ∈ On ↔ (∅ ↑𝑜 𝐵) ∈ On))
1613, 15syl5ibr 236 . . 3 (𝐴 = ∅ → (𝐵 ∈ On → (𝐴𝑜 𝐵) ∈ On))
1716impcom 394 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) ∈ On)
18 oveq2 6801 . . . . . . 7 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
1918eleq1d 2835 . . . . . 6 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 ∅) ∈ On))
20 oveq2 6801 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
2120eleq1d 2835 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 𝑦) ∈ On))
22 oveq2 6801 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
2322eleq1d 2835 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 suc 𝑦) ∈ On))
24 oveq2 6801 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
2524eleq1d 2835 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 𝐵) ∈ On))
26 oe0 7756 . . . . . . . 8 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
2726, 3syl6eqel 2858 . . . . . . 7 (𝐴 ∈ On → (𝐴𝑜 ∅) ∈ On)
2827adantr 466 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴𝑜 ∅) ∈ On)
29 omcl 7770 . . . . . . . . . . 11 (((𝐴𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ On)
3029expcom 398 . . . . . . . . . 10 (𝐴 ∈ On → ((𝐴𝑜 𝑦) ∈ On → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ On))
3130adantr 466 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝑦) ∈ On → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ On))
32 oesuc 7761 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
3332eleq1d 2835 . . . . . . . . 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 398 . . . . . . 7 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 suc 𝑦) ∈ On)))
3635adantrd 479 . . . . . 6 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 suc 𝑦) ∈ On)))
37 vex 3354 . . . . . . . . 9 𝑥 ∈ V
38 iunon 7589 . . . . . . . . 9 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On) → 𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
3937, 38mpan 670 . . . . . . . 8 (∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On → 𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
40 oelim 7768 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4137, 40mpanlr1 686 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4241anasss 457 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4342an12s 628 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4443eleq1d 2835 . . . . . . . 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 397 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 𝑥) ∈ On)))
4719, 21, 23, 25, 28, 36, 46tfinds3 7211 . . . . 5 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) ∈ On))
4847expd 400 . . . 4 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (𝐴𝑜 𝐵) ∈ On)))
4948com12 32 . . 3 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝐴𝑜 𝐵) ∈ On)))
5049imp31 404 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) ∈ On)
5117, 50oe0lem 7747 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  c0 4063   ciun 4654  Oncon0 5866  Lim wlim 5867  suc csuc 5868  (class class class)co 6793  1𝑜c1o 7706   ·𝑜 comu 7711  𝑜 coe 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-oexp 7719
This theorem is referenced by:  oen0  7820  oeordi  7821  oeord  7822  oecan  7823  oeword  7824  oewordri  7826  oeworde  7827  oeordsuc  7828  oeoalem  7830  oeoa  7831  oeoelem  7832  oeoe  7833  oelimcl  7834  oeeulem  7835  oeeui  7836  oaabs2  7879  omabs  7881  cantnfle  8732  cantnflt  8733  cantnfp1  8742  cantnflem1d  8749  cantnflem1  8750  cantnflem2  8751  cantnflem3  8752  cantnflem4  8753  cantnf  8754  oemapwe  8755  cantnffval2  8756  cnfcomlem  8760  cnfcom  8761  cnfcom3lem  8764  cnfcom3  8765  infxpenc  9041
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