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Theorem oeworde 7878
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 6850 . . . 4 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
31, 2sseq12d 3794 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ ∅ ⊆ (𝐴𝑜 ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 6850 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
64, 5sseq12d 3794 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝑦 ⊆ (𝐴𝑜 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 6850 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
97, 8sseq12d 3794 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 6850 . . . 4 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
1210, 11sseq12d 3794 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝐵 ⊆ (𝐴𝑜 𝐵)))
13 0ss 4134 . . . 4 ∅ ⊆ (𝐴𝑜 ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ⊆ (𝐴𝑜 ∅))
15 eloni 5918 . . . . . . 7 (𝑦 ∈ On → Ord 𝑦)
1615adantl 473 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord 𝑦)
17 eldifi 3894 . . . . . . . 8 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
18 oecl 7822 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
1917, 18sylan 575 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
20 eloni 5918 . . . . . . 7 ((𝐴𝑜 𝑦) ∈ On → Ord (𝐴𝑜 𝑦))
2119, 20syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord (𝐴𝑜 𝑦))
22 ordsucsssuc 7221 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴𝑜 𝑦)) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
2316, 21, 22syl2anc 579 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
24 suceloni 7211 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
25 oecl 7822 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) ∈ On)
2617, 24, 25syl2an 589 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) ∈ On)
27 eloni 5918 . . . . . . . 8 ((𝐴𝑜 suc 𝑦) ∈ On → Ord (𝐴𝑜 suc 𝑦))
2826, 27syl 17 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord (𝐴𝑜 suc 𝑦))
29 id 22 . . . . . . . 8 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ (On ∖ 2𝑜))
30 vex 3353 . . . . . . . . . 10 𝑦 ∈ V
3130sucid 5987 . . . . . . . . 9 𝑦 ∈ suc 𝑦
32 oeordi 7872 . . . . . . . . 9 ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑦 ∈ suc 𝑦 → (𝐴𝑜 𝑦) ∈ (𝐴𝑜 suc 𝑦)))
3331, 32mpi 20 . . . . . . . 8 ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐴𝑜 𝑦) ∈ (𝐴𝑜 suc 𝑦))
3424, 29, 33syl2anr 590 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ (𝐴𝑜 suc 𝑦))
35 ordsucss 7216 . . . . . . 7 (Ord (𝐴𝑜 suc 𝑦) → ((𝐴𝑜 𝑦) ∈ (𝐴𝑜 suc 𝑦) → suc (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
3628, 34, 35sylc 65 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → suc (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦))
37 sstr2 3768 . . . . . 6 (suc 𝑦 ⊆ suc (𝐴𝑜 𝑦) → (suc (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦) → suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦)))
3836, 37syl5com 31 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (suc 𝑦 ⊆ suc (𝐴𝑜 𝑦) → suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦)))
3923, 38sylbid 231 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴𝑜 𝑦) → suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦)))
4039expcom 402 . . 3 (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → (𝑦 ⊆ (𝐴𝑜 𝑦) → suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦))))
41 dif20el 7790 . . . . 5 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
4217, 41jca 507 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
43 ss2iun 4692 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴𝑜 𝑦))
44 limuni 5968 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
45 uniiun 4729 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4644, 45syl6eq 2815 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4746adantr 472 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
48 vex 3353 . . . . . . . . . 10 𝑥 ∈ V
49 oelim 7819 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5048, 49mpanlr1 697 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5150anasss 458 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5251an12s 639 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5347, 52sseq12d 3794 . . . . . 6 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝑦𝑥 𝑦 𝑦𝑥 (𝐴𝑜 𝑦)))
5443, 53syl5ibr 237 . . . . 5 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑥 ⊆ (𝐴𝑜 𝑥)))
5554ex 401 . . . 4 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑥 ⊆ (𝐴𝑜 𝑥))))
5642, 55syl5 34 . . 3 (Lim 𝑥 → (𝐴 ∈ (On ∖ 2𝑜) → (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑥 ⊆ (𝐴𝑜 𝑥))))
573, 6, 9, 12, 14, 40, 56tfinds3 7262 . 2 (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → 𝐵 ⊆ (𝐴𝑜 𝐵)))
5857impcom 396 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cdif 3729  wss 3732  c0 4079   cuni 4594   ciun 4676  Ord word 5907  Oncon0 5908  Lim wlim 5909  suc csuc 5910  (class class class)co 6842  2𝑜c2o 7758  𝑜 coe 7763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-omul 7769  df-oexp 7770
This theorem is referenced by:  oeeulem  7886  cnfcom3clem  8817
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