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Theorem oeworde 8631
Description: Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. Lemma 3.20 of [Schloeder] p. 10. (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 7439 . . . 4 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
31, 2sseq12d 4017 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴o 𝑥) ↔ ∅ ⊆ (𝐴o ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 7439 . . . 4 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
64, 5sseq12d 4017 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝑦 ⊆ (𝐴o 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 7439 . . . 4 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
97, 8sseq12d 4017 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ suc 𝑦 ⊆ (𝐴o suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 7439 . . . 4 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
1210, 11sseq12d 4017 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝐵 ⊆ (𝐴o 𝐵)))
13 0ss 4400 . . . 4 ∅ ⊆ (𝐴o ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2o) → ∅ ⊆ (𝐴o ∅))
15 eloni 6394 . . . . . 6 (𝑦 ∈ On → Ord 𝑦)
16 eldifi 4131 . . . . . . . 8 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
17 oecl 8575 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
1816, 17sylan 580 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
19 eloni 6394 . . . . . . 7 ((𝐴o 𝑦) ∈ On → Ord (𝐴o 𝑦))
2018, 19syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴o 𝑦))
21 ordsucsssuc 7843 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴o 𝑦)) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
2215, 20, 21syl2an2 686 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
23 onsuc 7831 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
24 oecl 8575 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴o suc 𝑦) ∈ On)
2516, 23, 24syl2an 596 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) ∈ On)
26 eloni 6394 . . . . . . . 8 ((𝐴o suc 𝑦) ∈ On → Ord (𝐴o suc 𝑦))
2725, 26syl 17 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴o suc 𝑦))
28 id 22 . . . . . . . 8 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ (On ∖ 2o))
29 vex 3484 . . . . . . . . . 10 𝑦 ∈ V
3029sucid 6466 . . . . . . . . 9 𝑦 ∈ suc 𝑦
31 oeordi 8625 . . . . . . . . 9 ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦 ∈ suc 𝑦 → (𝐴o 𝑦) ∈ (𝐴o suc 𝑦)))
3230, 31mpi 20 . . . . . . . 8 ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐴o 𝑦) ∈ (𝐴o suc 𝑦))
3323, 28, 32syl2anr 597 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ (𝐴o suc 𝑦))
34 ordsucss 7838 . . . . . . 7 (Ord (𝐴o suc 𝑦) → ((𝐴o 𝑦) ∈ (𝐴o suc 𝑦) → suc (𝐴o 𝑦) ⊆ (𝐴o suc 𝑦)))
3527, 33, 34sylc 65 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → suc (𝐴o 𝑦) ⊆ (𝐴o suc 𝑦))
36 sstr2 3990 . . . . . 6 (suc 𝑦 ⊆ suc (𝐴o 𝑦) → (suc (𝐴o 𝑦) ⊆ (𝐴o suc 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦)))
3735, 36syl5com 31 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (suc 𝑦 ⊆ suc (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦)))
3822, 37sylbid 240 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦)))
3938expcom 413 . . 3 (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2o) → (𝑦 ⊆ (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦))))
40 dif20el 8543 . . . . 5 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
4116, 40jca 511 . . . 4 (𝐴 ∈ (On ∖ 2o) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
42 ss2iun 5010 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴o 𝑦))
43 limuni 6445 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
44 uniiun 5058 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4543, 44eqtrdi 2793 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4645adantr 480 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
47 vex 3484 . . . . . . . . . 10 𝑥 ∈ V
48 oelim 8572 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4947, 48mpanlr1 706 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5049anasss 466 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5150an12s 649 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5246, 51sseq12d 4017 . . . . . 6 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝑦𝑥 𝑦 𝑦𝑥 (𝐴o 𝑦)))
5342, 52imbitrrid 246 . . . . 5 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑥 ⊆ (𝐴o 𝑥)))
5453ex 412 . . . 4 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑥 ⊆ (𝐴o 𝑥))))
5541, 54syl5 34 . . 3 (Lim 𝑥 → (𝐴 ∈ (On ∖ 2o) → (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑥 ⊆ (𝐴o 𝑥))))
563, 6, 9, 12, 14, 39, 55tfinds3 7886 . 2 (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2o) → 𝐵 ⊆ (𝐴o 𝐵)))
5756impcom 407 1 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cdif 3948  wss 3951  c0 4333   cuni 4907   ciun 4991  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431  2oc2o 8500  o coe 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512
This theorem is referenced by:  oeeulem  8639  cnfcom3clem  9745  oege2  43320  nnoeomeqom  43325
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