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Theorem oeworde 8540
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 7365 . . . 4 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
31, 2sseq12d 3977 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴o 𝑥) ↔ ∅ ⊆ (𝐴o ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 7365 . . . 4 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
64, 5sseq12d 3977 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝑦 ⊆ (𝐴o 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 7365 . . . 4 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
97, 8sseq12d 3977 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ suc 𝑦 ⊆ (𝐴o suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 7365 . . . 4 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
1210, 11sseq12d 3977 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝐵 ⊆ (𝐴o 𝐵)))
13 0ss 4356 . . . 4 ∅ ⊆ (𝐴o ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2o) → ∅ ⊆ (𝐴o ∅))
15 eloni 6327 . . . . . 6 (𝑦 ∈ On → Ord 𝑦)
16 eldifi 4086 . . . . . . . 8 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
17 oecl 8483 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
1816, 17sylan 580 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
19 eloni 6327 . . . . . . 7 ((𝐴o 𝑦) ∈ On → Ord (𝐴o 𝑦))
2018, 19syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴o 𝑦))
21 ordsucsssuc 7758 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴o 𝑦)) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
2215, 20, 21syl2an2 684 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
23 onsuc 7746 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
24 oecl 8483 . . . . . . . . 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 6327 . . . . . . . 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 3449 . . . . . . . . . 10 𝑦 ∈ V
3029sucid 6399 . . . . . . . . 9 𝑦 ∈ suc 𝑦
31 oeordi 8534 . . . . . . . . 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 7753 . . . . . . 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 3951 . . . . . 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 239 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦)))
3938expcom 414 . . 3 (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2o) → (𝑦 ⊆ (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦))))
40 dif20el 8451 . . . . 5 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
4116, 40jca 512 . . . 4 (𝐴 ∈ (On ∖ 2o) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
42 ss2iun 4972 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴o 𝑦))
43 limuni 6378 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
44 uniiun 5018 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4543, 44eqtrdi 2792 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4645adantr 481 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
47 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
48 oelim 8480 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4947, 48mpanlr1 704 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5049anasss 467 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5150an12s 647 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5246, 51sseq12d 3977 . . . . . 6 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝑦𝑥 𝑦 𝑦𝑥 (𝐴o 𝑦)))
5342, 52syl5ibr 245 . . . . 5 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑥 ⊆ (𝐴o 𝑥)))
5453ex 413 . . . 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 7801 . 2 (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2o) → 𝐵 ⊆ (𝐴o 𝐵)))
5756impcom 408 1 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cdif 3907  wss 3910  c0 4282   cuni 4865   ciun 4954  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357  2oc2o 8406  o coe 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-oexp 8418
This theorem is referenced by:  oeeulem  8548  cnfcom3clem  9641  oege2  41627  nnoeomeqom  41632
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