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Theorem oeworde 7560
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 6557 . . . 4 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
31, 2sseq12d 3597 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ ∅ ⊆ (𝐴𝑜 ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 6557 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
64, 5sseq12d 3597 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝑦 ⊆ (𝐴𝑜 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 6557 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
97, 8sseq12d 3597 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 6557 . . . 4 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
1210, 11sseq12d 3597 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝐵 ⊆ (𝐴𝑜 𝐵)))
13 0ss 3924 . . . 4 ∅ ⊆ (𝐴𝑜 ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ⊆ (𝐴𝑜 ∅))
15 eloni 5650 . . . . . . 7 (𝑦 ∈ On → Ord 𝑦)
1615adantl 481 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord 𝑦)
17 eldifi 3694 . . . . . . . 8 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
18 oecl 7504 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
1917, 18sylan 487 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
20 eloni 5650 . . . . . . 7 ((𝐴𝑜 𝑦) ∈ On → Ord (𝐴𝑜 𝑦))
2119, 20syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord (𝐴𝑜 𝑦))
22 ordsucsssuc 6915 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴𝑜 𝑦)) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
2316, 21, 22syl2anc 691 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
24 suceloni 6905 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
25 oecl 7504 . . . . . . . . 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 5650 . . . . . . . 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 3176 . . . . . . . . . 10 𝑦 ∈ V
3130sucid 5721 . . . . . . . . 9 𝑦 ∈ suc 𝑦
32 oeordi 7554 . . . . . . . . 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 6910 . . . . . . 7 (Ord (𝐴𝑜 suc 𝑦) → ((𝐴𝑜 𝑦) ∈ (𝐴𝑜 suc 𝑦) → suc (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
3628, 34, 35sylc 63 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → suc (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦))
37 sstr2 3575 . . . . . 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 229 . . . 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 7472 . . . . 5 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
4217, 41jca 553 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
43 ss2iun 4472 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴𝑜 𝑦))
44 limuni 5702 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
45 uniiun 4509 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4644, 45syl6eq 2660 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4746adantr 480 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
48 vex 3176 . . . . . . . . . 10 𝑥 ∈ V
49 oelim 7501 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5048, 49mpanlr1 718 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5150anasss 677 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5251an12s 839 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5347, 52sseq12d 3597 . . . . . 6 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝑦𝑥 𝑦 𝑦𝑥 (𝐴𝑜 𝑦)))
5443, 53syl5ibr 235 . . . . 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 33 . . 3 (Lim 𝑥 → (𝐴 ∈ (On ∖ 2𝑜) → (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑥 ⊆ (𝐴𝑜 𝑥))))
573, 6, 9, 12, 14, 40, 56tfinds3 6956 . 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 195  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cdif 3537  wss 3540  c0 3874   cuni 4372   ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  2𝑜c2o 7441  𝑜 coe 7446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453
This theorem is referenced by:  oeeulem  7568  cnfcom3clem  8485
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