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Theorem oecl 8153
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 7156 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2 oe0m0 8136 . . . . . . . . 9 (∅ ↑o ∅) = 1o
3 1on 8100 . . . . . . . . 9 1o ∈ On
42, 3eqeltri 2914 . . . . . . . 8 (∅ ↑o ∅) ∈ On
51, 4syl6eqel 2926 . . . . . . 7 (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
65adantl 482 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
7 oe0m1 8137 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
87biimpa 477 . . . . . . . 8 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
9 0elon 6242 . . . . . . . 8 ∅ ∈ On
108, 9syl6eqel 2926 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
1110adantll 710 . . . . . 6 (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
126, 11oe0lem 8129 . . . . 5 ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
1312anidms 567 . . . 4 (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
14 oveq1 7155 . . . . 5 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
1514eleq1d 2902 . . . 4 (𝐴 = ∅ → ((𝐴o 𝐵) ∈ On ↔ (∅ ↑o 𝐵) ∈ On))
1613, 15syl5ibr 247 . . 3 (𝐴 = ∅ → (𝐵 ∈ On → (𝐴o 𝐵) ∈ On))
1716impcom 408 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴o 𝐵) ∈ On)
18 oveq2 7156 . . . . . . 7 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
1918eleq1d 2902 . . . . . 6 (𝑥 = ∅ → ((𝐴o 𝑥) ∈ On ↔ (𝐴o ∅) ∈ On))
20 oveq2 7156 . . . . . . 7 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
2120eleq1d 2902 . . . . . 6 (𝑥 = 𝑦 → ((𝐴o 𝑥) ∈ On ↔ (𝐴o 𝑦) ∈ On))
22 oveq2 7156 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
2322eleq1d 2902 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ∈ On ↔ (𝐴o suc 𝑦) ∈ On))
24 oveq2 7156 . . . . . . 7 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
2524eleq1d 2902 . . . . . 6 (𝑥 = 𝐵 → ((𝐴o 𝑥) ∈ On ↔ (𝐴o 𝐵) ∈ On))
26 oe0 8138 . . . . . . . 8 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2726, 3syl6eqel 2926 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) ∈ On)
2827adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o ∅) ∈ On)
29 omcl 8152 . . . . . . . . . . 11 (((𝐴o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴o 𝑦) ·o 𝐴) ∈ On)
3029expcom 414 . . . . . . . . . 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 8143 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
3332eleq1d 2902 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o suc 𝑦) ∈ On ↔ ((𝐴o 𝑦) ·o 𝐴) ∈ On))
3431, 33sylibrd 260 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝑦) ∈ On → (𝐴o suc 𝑦) ∈ On))
3534expcom 414 . . . . . . 7 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴o 𝑦) ∈ On → (𝐴o suc 𝑦) ∈ On)))
3635adantrd 492 . . . . . 6 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴o 𝑦) ∈ On → (𝐴o suc 𝑦) ∈ On)))
37 vex 3503 . . . . . . . . 9 𝑥 ∈ V
38 iunon 7967 . . . . . . . . 9 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴o 𝑦) ∈ On) → 𝑦𝑥 (𝐴o 𝑦) ∈ On)
3937, 38mpan 686 . . . . . . . 8 (∀𝑦𝑥 (𝐴o 𝑦) ∈ On → 𝑦𝑥 (𝐴o 𝑦) ∈ On)
40 oelim 8150 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4137, 40mpanlr1 702 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4241anasss 467 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4342an12s 645 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4443eleq1d 2902 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴o 𝑥) ∈ On ↔ 𝑦𝑥 (𝐴o 𝑦) ∈ On))
4539, 44syl5ibr 247 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦𝑥 (𝐴o 𝑦) ∈ On → (𝐴o 𝑥) ∈ On))
4645ex 413 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 (𝐴o 𝑦) ∈ On → (𝐴o 𝑥) ∈ On)))
4719, 21, 23, 25, 28, 36, 46tfinds3 7567 . . . . 5 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) ∈ On))
4847expd 416 . . . 4 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (𝐴o 𝐵) ∈ On)))
4948com12 32 . . 3 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝐴o 𝐵) ∈ On)))
5049imp31 418 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) ∈ On)
5117, 50oe0lem 8129 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  c0 4295   ciun 4917  Oncon0 6189  Lim wlim 6190  suc csuc 6191  (class class class)co 7148  1oc1o 8086   ·o comu 8091  o coe 8092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-omul 8098  df-oexp 8099
This theorem is referenced by:  oen0  8202  oeordi  8203  oeord  8204  oecan  8205  oeword  8206  oewordri  8208  oeworde  8209  oeordsuc  8210  oeoalem  8212  oeoa  8213  oeoelem  8214  oeoe  8215  oelimcl  8216  oeeulem  8217  oeeui  8218  oaabs2  8262  omabs  8264  cantnfle  9123  cantnflt  9124  cantnfp1  9133  cantnflem1d  9140  cantnflem1  9141  cantnflem2  9142  cantnflem3  9143  cantnflem4  9144  cantnf  9145  oemapwe  9146  cantnffval2  9147  cnfcomlem  9151  cnfcom  9152  cnfcom3lem  9155  cnfcom3  9156  infxpenc  9433
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