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Theorem oeworde 7827
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 6801 . . . 4 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
31, 2sseq12d 3783 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ ∅ ⊆ (𝐴𝑜 ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 6801 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
64, 5sseq12d 3783 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝑦 ⊆ (𝐴𝑜 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 6801 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
97, 8sseq12d 3783 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 6801 . . . 4 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
1210, 11sseq12d 3783 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴𝑜 𝑥) ↔ 𝐵 ⊆ (𝐴𝑜 𝐵)))
13 0ss 4116 . . . 4 ∅ ⊆ (𝐴𝑜 ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ⊆ (𝐴𝑜 ∅))
15 eloni 5876 . . . . . . 7 (𝑦 ∈ On → Ord 𝑦)
1615adantl 467 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord 𝑦)
17 eldifi 3883 . . . . . . . 8 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
18 oecl 7771 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
1917, 18sylan 569 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
20 eloni 5876 . . . . . . 7 ((𝐴𝑜 𝑦) ∈ On → Ord (𝐴𝑜 𝑦))
2119, 20syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord (𝐴𝑜 𝑦))
22 ordsucsssuc 7170 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴𝑜 𝑦)) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
2316, 21, 22syl2anc 573 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴𝑜 𝑦)))
24 suceloni 7160 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
25 oecl 7771 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) ∈ On)
2617, 24, 25syl2an 583 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) ∈ On)
27 eloni 5876 . . . . . . . 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 3354 . . . . . . . . . 10 𝑦 ∈ V
3130sucid 5947 . . . . . . . . 9 𝑦 ∈ suc 𝑦
32 oeordi 7821 . . . . . . . . 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 584 . . . . . . 7 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ (𝐴𝑜 suc 𝑦))
35 ordsucss 7165 . . . . . . 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 3759 . . . . . 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 398 . . 3 (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → (𝑦 ⊆ (𝐴𝑜 𝑦) → suc 𝑦 ⊆ (𝐴𝑜 suc 𝑦))))
41 dif20el 7739 . . . . 5 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
4217, 41jca 501 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
43 ss2iun 4670 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴𝑜 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴𝑜 𝑦))
44 limuni 5928 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
45 uniiun 4707 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4644, 45syl6eq 2821 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4746adantr 466 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
48 vex 3354 . . . . . . . . . 10 𝑥 ∈ V
49 oelim 7768 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5048, 49mpanlr1 686 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5150anasss 457 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5251an12s 628 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
5347, 52sseq12d 3783 . . . . . 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 397 . . . 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 7211 . 2 (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → 𝐵 ⊆ (𝐴𝑜 𝐵)))
5857impcom 394 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cdif 3720  wss 3723  c0 4063   cuni 4574   ciun 4654  Ord word 5865  Oncon0 5866  Lim wlim 5867  suc csuc 5868  (class class class)co 6793  2𝑜c2o 7707  𝑜 coe 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-oexp 7719
This theorem is referenced by:  oeeulem  7835  cnfcom3clem  8766
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