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Theorem oecl 7569
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 6618 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2 oe0m0 7552 . . . . . . . . 9 (∅ ↑𝑜 ∅) = 1𝑜
3 1on 7519 . . . . . . . . 9 1𝑜 ∈ On
42, 3eqeltri 2694 . . . . . . . 8 (∅ ↑𝑜 ∅) ∈ On
51, 4syl6eqel 2706 . . . . . . 7 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) ∈ On)
65adantl 482 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑𝑜 𝐵) ∈ On)
7 oe0m1 7553 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
87biimpa 501 . . . . . . . 8 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
9 0elon 5742 . . . . . . . 8 ∅ ∈ On
108, 9syl6eqel 2706 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
1110adantll 749 . . . . . 6 (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
126, 11oe0lem 7545 . . . . 5 ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
1312anidms 676 . . . 4 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
14 oveq1 6617 . . . . 5 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
1514eleq1d 2683 . . . 4 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ∈ On ↔ (∅ ↑𝑜 𝐵) ∈ On))
1613, 15syl5ibr 236 . . 3 (𝐴 = ∅ → (𝐵 ∈ On → (𝐴𝑜 𝐵) ∈ On))
1716impcom 446 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) ∈ On)
18 oveq2 6618 . . . . . . 7 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
1918eleq1d 2683 . . . . . 6 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 ∅) ∈ On))
20 oveq2 6618 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
2120eleq1d 2683 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 𝑦) ∈ On))
22 oveq2 6618 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
2322eleq1d 2683 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 suc 𝑦) ∈ On))
24 oveq2 6618 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
2524eleq1d 2683 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑜 𝑥) ∈ On ↔ (𝐴𝑜 𝐵) ∈ On))
26 oe0 7554 . . . . . . . 8 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
2726, 3syl6eqel 2706 . . . . . . 7 (𝐴 ∈ On → (𝐴𝑜 ∅) ∈ On)
2827adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴𝑜 ∅) ∈ On)
29 omcl 7568 . . . . . . . . . . 11 (((𝐴𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ On)
3029expcom 451 . . . . . . . . . 10 (𝐴 ∈ On → ((𝐴𝑜 𝑦) ∈ On → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ On))
3130adantr 481 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝑦) ∈ On → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ On))
32 oesuc 7559 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
3332eleq1d 2683 . . . . . . . . 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 451 . . . . . . 7 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 suc 𝑦) ∈ On)))
3635adantrd 484 . . . . . 6 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 suc 𝑦) ∈ On)))
37 vex 3192 . . . . . . . . 9 𝑥 ∈ V
38 iunon 7388 . . . . . . . . 9 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On) → 𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
3937, 38mpan 705 . . . . . . . 8 (∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On → 𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
40 oelim 7566 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4137, 40mpanlr1 721 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4241anasss 678 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4342an12s 842 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4443eleq1d 2683 . . . . . . . 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 450 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On → (𝐴𝑜 𝑥) ∈ On)))
4719, 21, 23, 25, 28, 36, 46tfinds3 7018 . . . . 5 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) ∈ On))
4847expd 452 . . . 4 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (𝐴𝑜 𝐵) ∈ On)))
4948com12 32 . . 3 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝐴𝑜 𝐵) ∈ On)))
5049imp31 448 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) ∈ On)
5117, 50oe0lem 7545 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  c0 3896   ciun 4490  Oncon0 5687  Lim wlim 5688  suc csuc 5689  (class class class)co 6610  1𝑜c1o 7505   ·𝑜 comu 7510  𝑜 coe 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-omul 7517  df-oexp 7518
This theorem is referenced by:  oen0  7618  oeordi  7619  oeord  7620  oecan  7621  oeword  7622  oewordri  7624  oeworde  7625  oeordsuc  7626  oeoalem  7628  oeoa  7629  oeoelem  7630  oeoe  7631  oelimcl  7632  oeeulem  7633  oeeui  7634  oaabs2  7677  omabs  7679  cantnfle  8519  cantnflt  8520  cantnfp1  8529  cantnflem1d  8536  cantnflem1  8537  cantnflem2  8538  cantnflem3  8539  cantnflem4  8540  cantnf  8541  oemapwe  8542  cantnffval2  8543  cnfcomlem  8547  cnfcom  8548  cnfcom3lem  8551  cnfcom3  8552  infxpenc  8792
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