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Theorem oeworde 7618
 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 6612 . . . 4 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
31, 2sseq12d 3613 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ ∅ ⊆ (𝐴𝑜 ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 6612 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
64, 5sseq12d 3613 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝑦 ⊆ (𝐴𝑜 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 6612 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
97, 8sseq12d 3613 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 6612 . . . 4 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
1210, 11sseq12d 3613 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝐵 ⊆ (𝐴𝑜 𝐵)))
13 0ss 3944 . . . 4 ∅ ⊆ (𝐴𝑜 ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ⊆ (𝐴𝑜 ∅))
15 eloni 5692 . . . . . . 7 (𝑦 ∈ On → Ord 𝑦)
1615adantl 482 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord 𝑦)
17 eldifi 3710 . . . . . . . 8 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
18 oecl 7562 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
1917, 18sylan 488 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
20 eloni 5692 . . . . . . 7 ((𝐴𝑜 𝑦) ∈ On → Ord (𝐴𝑜 𝑦))
2119, 20syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord (𝐴𝑜 𝑦))
22 ordsucsssuc 6970 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴𝑜 𝑦)) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
2316, 21, 22syl2anc 692 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
24 suceloni 6960 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
25 oecl 7562 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) ∈ On)
2617, 24, 25syl2an 494 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) ∈ On)
27 eloni 5692 . . . . . . . 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 3189 . . . . . . . . . 10 𝑦 ∈ V
3130sucid 5763 . . . . . . . . 9 𝑦 ∈ suc 𝑦
32 oeordi 7612 . . . . . . . . 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 495 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ (𝐴𝑜 suc 𝑦))
35 ordsucss 6965 . . . . . . 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 3590 . . . . . 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 451 . . 3 (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → (𝑦 ⊆ (𝐴𝑜 𝑦) → suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦))))
41 dif20el 7530 . . . . 5 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
4217, 41jca 554 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
43 ss2iun 4502 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴𝑜 𝑦))
44 limuni 5744 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
45 uniiun 4539 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4644, 45syl6eq 2671 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4746adantr 481 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
48 vex 3189 . . . . . . . . . 10 𝑥 ∈ V
49 oelim 7559 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5048, 49mpanlr1 721 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5150anasss 678 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5251an12s 842 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5347, 52sseq12d 3613 . . . . . 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 450 . . . 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 7011 . 2 (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → 𝐵 ⊆ (𝐴𝑜 𝐵)))
5857impcom 446 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴𝑜 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   ∖ cdif 3552   ⊆ wss 3555  ∅c0 3891  ∪ cuni 4402  ∪ ciun 4485  Ord word 5681  Oncon0 5682  Lim wlim 5683  suc csuc 5684  (class class class)co 6604  2𝑜c2o 7499   ↑𝑜 coe 7504 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511 This theorem is referenced by:  oeeulem  7626  cnfcom3clem  8546
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