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Theorem oeworde 8214
Description: Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oeworde ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))

Proof of Theorem oeworde
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝑥 = ∅ → 𝑥 = ∅)
2 oveq2 7158 . . . 4 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
31, 2sseq12d 4004 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴o 𝑥) ↔ ∅ ⊆ (𝐴o ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 7158 . . . 4 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
64, 5sseq12d 4004 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝑦 ⊆ (𝐴o 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 7158 . . . 4 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
97, 8sseq12d 4004 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ suc 𝑦 ⊆ (𝐴o suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 7158 . . . 4 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
1210, 11sseq12d 4004 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝐵 ⊆ (𝐴o 𝐵)))
13 0ss 4354 . . . 4 ∅ ⊆ (𝐴o ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2o) → ∅ ⊆ (𝐴o ∅))
15 eloni 6200 . . . . . 6 (𝑦 ∈ On → Ord 𝑦)
16 eldifi 4107 . . . . . . . 8 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
17 oecl 8158 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
1816, 17sylan 580 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
19 eloni 6200 . . . . . . 7 ((𝐴o 𝑦) ∈ On → Ord (𝐴o 𝑦))
2018, 19syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴o 𝑦))
21 ordsucsssuc 7531 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴o 𝑦)) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
2215, 20, 21syl2an2 682 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
23 suceloni 7521 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
24 oecl 8158 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴o suc 𝑦) ∈ On)
2516, 23, 24syl2an 595 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) ∈ On)
26 eloni 6200 . . . . . . . 8 ((𝐴o suc 𝑦) ∈ On → Ord (𝐴o suc 𝑦))
2725, 26syl 17 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴o suc 𝑦))
28 id 22 . . . . . . . 8 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ (On ∖ 2o))
29 vex 3503 . . . . . . . . . 10 𝑦 ∈ V
3029sucid 6269 . . . . . . . . 9 𝑦 ∈ suc 𝑦
31 oeordi 8208 . . . . . . . . 9 ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦 ∈ suc 𝑦 → (𝐴o 𝑦) ∈ (𝐴o suc 𝑦)))
3230, 31mpi 20 . . . . . . . 8 ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐴o 𝑦) ∈ (𝐴o suc 𝑦))
3323, 28, 32syl2anr 596 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ (𝐴o suc 𝑦))
34 ordsucss 7526 . . . . . . 7 (Ord (𝐴o suc 𝑦) → ((𝐴o 𝑦) ∈ (𝐴o suc 𝑦) → suc (𝐴o 𝑦) ⊆ (𝐴o suc 𝑦)))
3527, 33, 34sylc 65 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → suc (𝐴o 𝑦) ⊆ (𝐴o suc 𝑦))
36 sstr2 3978 . . . . . 6 (suc 𝑦 ⊆ suc (𝐴o 𝑦) → (suc (𝐴o 𝑦) ⊆ (𝐴o suc 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦)))
3735, 36syl5com 31 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (suc 𝑦 ⊆ suc (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦)))
3822, 37sylbid 241 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦)))
3938expcom 414 . . 3 (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2o) → (𝑦 ⊆ (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦))))
40 dif20el 8126 . . . . 5 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
4116, 40jca 512 . . . 4 (𝐴 ∈ (On ∖ 2o) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
42 ss2iun 4934 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴o 𝑦))
43 limuni 6250 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
44 uniiun 4979 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4543, 44syl6eq 2877 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4645adantr 481 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
47 vex 3503 . . . . . . . . . 10 𝑥 ∈ V
48 oelim 8155 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4947, 48mpanlr1 702 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5049anasss 467 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5150an12s 645 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5246, 51sseq12d 4004 . . . . . 6 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝑦𝑥 𝑦 𝑦𝑥 (𝐴o 𝑦)))
5342, 52syl5ibr 247 . . . . 5 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑥 ⊆ (𝐴o 𝑥)))
5453ex 413 . . . 4 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑥 ⊆ (𝐴o 𝑥))))
5541, 54syl5 34 . . 3 (Lim 𝑥 → (𝐴 ∈ (On ∖ 2o) → (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑥 ⊆ (𝐴o 𝑥))))
563, 6, 9, 12, 14, 39, 55tfinds3 7572 . 2 (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2o) → 𝐵 ⊆ (𝐴o 𝐵)))
5756impcom 408 1 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  cdif 3937  wss 3940  c0 4295   cuni 4837   ciun 4917  Ord word 6189  Oncon0 6190  Lim wlim 6191  suc csuc 6192  (class class class)co 7150  2oc2o 8092  o coe 8097
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 7455
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 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-omul 8103  df-oexp 8104
This theorem is referenced by:  oeeulem  8222  cnfcom3clem  9162
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