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Theorem oeworde 8596
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 7420 . . . 4 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
31, 2sseq12d 4015 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴o 𝑥) ↔ ∅ ⊆ (𝐴o ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 7420 . . . 4 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
64, 5sseq12d 4015 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝑦 ⊆ (𝐴o 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 7420 . . . 4 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
97, 8sseq12d 4015 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ suc 𝑦 ⊆ (𝐴o suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 7420 . . . 4 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
1210, 11sseq12d 4015 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝐵 ⊆ (𝐴o 𝐵)))
13 0ss 4396 . . . 4 ∅ ⊆ (𝐴o ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2o) → ∅ ⊆ (𝐴o ∅))
15 eloni 6374 . . . . . 6 (𝑦 ∈ On → Ord 𝑦)
16 eldifi 4126 . . . . . . . 8 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
17 oecl 8540 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
1816, 17sylan 579 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
19 eloni 6374 . . . . . . 7 ((𝐴o 𝑦) ∈ On → Ord (𝐴o 𝑦))
2018, 19syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴o 𝑦))
21 ordsucsssuc 7814 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴o 𝑦)) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
2215, 20, 21syl2an2 683 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
23 onsuc 7802 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
24 oecl 8540 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴o suc 𝑦) ∈ On)
2516, 23, 24syl2an 595 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) ∈ On)
26 eloni 6374 . . . . . . . 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 3477 . . . . . . . . . 10 𝑦 ∈ V
3029sucid 6446 . . . . . . . . 9 𝑦 ∈ suc 𝑦
31 oeordi 8590 . . . . . . . . 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 596 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ (𝐴o suc 𝑦))
34 ordsucss 7809 . . . . . . 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 3989 . . . . . 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 413 . . 3 (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2o) → (𝑦 ⊆ (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦))))
40 dif20el 8508 . . . . 5 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
4116, 40jca 511 . . . 4 (𝐴 ∈ (On ∖ 2o) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
42 ss2iun 5015 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴o 𝑦))
43 limuni 6425 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
44 uniiun 5061 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4543, 44eqtrdi 2787 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4645adantr 480 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
47 vex 3477 . . . . . . . . . 10 𝑥 ∈ V
48 oelim 8537 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4947, 48mpanlr1 703 . . . . . . . . 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 646 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5246, 51sseq12d 4015 . . . . . 6 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝑦𝑥 𝑦 𝑦𝑥 (𝐴o 𝑦)))
5342, 52imbitrrid 245 . . . . 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 7857 . 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 205  wa 395   = wceq 1540  wcel 2105  wral 3060  Vcvv 3473  cdif 3945  wss 3948  c0 4322   cuni 4908   ciun 4997  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7412  2oc2o 8463  o coe 8468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-oadd 8473  df-omul 8474  df-oexp 8475
This theorem is referenced by:  oeeulem  8604  cnfcom3clem  9703  oege2  42360  nnoeomeqom  42365
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