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Theorem oeworde 7718
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 6698 . . . 4 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
31, 2sseq12d 3667 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ ∅ ⊆ (𝐴𝑜 ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 6698 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
64, 5sseq12d 3667 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝑦 ⊆ (𝐴𝑜 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 6698 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
97, 8sseq12d 3667 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 6698 . . . 4 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
1210, 11sseq12d 3667 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝐵 ⊆ (𝐴𝑜 𝐵)))
13 0ss 4005 . . . 4 ∅ ⊆ (𝐴𝑜 ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ⊆ (𝐴𝑜 ∅))
15 eloni 5771 . . . . . . 7 (𝑦 ∈ On → Ord 𝑦)
1615adantl 481 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord 𝑦)
17 eldifi 3765 . . . . . . . 8 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
18 oecl 7662 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
1917, 18sylan 487 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
20 eloni 5771 . . . . . . 7 ((𝐴𝑜 𝑦) ∈ On → Ord (𝐴𝑜 𝑦))
2119, 20syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord (𝐴𝑜 𝑦))
22 ordsucsssuc 7065 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴𝑜 𝑦)) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
2316, 21, 22syl2anc 694 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
24 suceloni 7055 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
25 oecl 7662 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) ∈ On)
2617, 24, 25syl2an 493 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) ∈ On)
27 eloni 5771 . . . . . . . 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 3234 . . . . . . . . . 10 𝑦 ∈ V
3130sucid 5842 . . . . . . . . 9 𝑦 ∈ suc 𝑦
32 oeordi 7712 . . . . . . . . 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 494 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ (𝐴𝑜 suc 𝑦))
35 ordsucss 7060 . . . . . . 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 3643 . . . . . 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 230 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴𝑜 𝑦) → suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦)))
4039expcom 450 . . 3 (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → (𝑦 ⊆ (𝐴𝑜 𝑦) → suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦))))
41 dif20el 7630 . . . . 5 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
4217, 41jca 553 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
43 ss2iun 4568 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴𝑜 𝑦))
44 limuni 5823 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
45 uniiun 4605 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4644, 45syl6eq 2701 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4746adantr 480 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
48 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
49 oelim 7659 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5048, 49mpanlr1 722 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5150anasss 680 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5251an12s 860 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5347, 52sseq12d 3667 . . . . . 6 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝑦𝑥 𝑦 𝑦𝑥 (𝐴𝑜 𝑦)))
5443, 53syl5ibr 236 . . . . 5 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑥 ⊆ (𝐴𝑜 𝑥)))
5554ex 449 . . . 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 7106 . 2 (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → 𝐵 ⊆ (𝐴𝑜 𝐵)))
5857impcom 445 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  wss 3607  c0 3948   cuni 4468   ciun 4552  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690  2𝑜c2o 7599  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by:  oeeulem  7726  cnfcom3clem  8640
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