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Theorem oeworde 8593
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 7417 . . . 4 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
31, 2sseq12d 4016 . . 3 (𝑥 = ∅ → (𝑥 ⊆ (𝐴o 𝑥) ↔ ∅ ⊆ (𝐴o ∅)))
4 id 22 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 oveq2 7417 . . . 4 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
64, 5sseq12d 4016 . . 3 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝑦 ⊆ (𝐴o 𝑦)))
7 id 22 . . . 4 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
8 oveq2 7417 . . . 4 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
97, 8sseq12d 4016 . . 3 (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴o 𝑥) ↔ suc 𝑦 ⊆ (𝐴o suc 𝑦)))
10 id 22 . . . 4 (𝑥 = 𝐵𝑥 = 𝐵)
11 oveq2 7417 . . . 4 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
1210, 11sseq12d 4016 . . 3 (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴o 𝑥) ↔ 𝐵 ⊆ (𝐴o 𝐵)))
13 0ss 4397 . . . 4 ∅ ⊆ (𝐴o ∅)
1413a1i 11 . . 3 (𝐴 ∈ (On ∖ 2o) → ∅ ⊆ (𝐴o ∅))
15 eloni 6375 . . . . . 6 (𝑦 ∈ On → Ord 𝑦)
16 eldifi 4127 . . . . . . . 8 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
17 oecl 8537 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
1816, 17sylan 581 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
19 eloni 6375 . . . . . . 7 ((𝐴o 𝑦) ∈ On → Ord (𝐴o 𝑦))
2018, 19syl 17 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴o 𝑦))
21 ordsucsssuc 7811 . . . . . 6 ((Ord 𝑦 ∧ Ord (𝐴o 𝑦)) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
2215, 20, 21syl2an2 685 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴o 𝑦)))
23 onsuc 7799 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
24 oecl 8537 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴o suc 𝑦) ∈ On)
2516, 23, 24syl2an 597 . . . . . . . 8 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) ∈ On)
26 eloni 6375 . . . . . . . 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 3479 . . . . . . . . . 10 𝑦 ∈ V
3029sucid 6447 . . . . . . . . 9 𝑦 ∈ suc 𝑦
31 oeordi 8587 . . . . . . . . 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 598 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ (𝐴o suc 𝑦))
34 ordsucss 7806 . . . . . . 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 239 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦)))
3938expcom 415 . . 3 (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2o) → (𝑦 ⊆ (𝐴o 𝑦) → suc 𝑦 ⊆ (𝐴o suc 𝑦))))
40 dif20el 8505 . . . . 5 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
4116, 40jca 513 . . . 4 (𝐴 ∈ (On ∖ 2o) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
42 ss2iun 5016 . . . . . 6 (∀𝑦𝑥 𝑦 ⊆ (𝐴o 𝑦) → 𝑦𝑥 𝑦 𝑦𝑥 (𝐴o 𝑦))
43 limuni 6426 . . . . . . . . 9 (Lim 𝑥𝑥 = 𝑥)
44 uniiun 5062 . . . . . . . . 9 𝑥 = 𝑦𝑥 𝑦
4543, 44eqtrdi 2789 . . . . . . . 8 (Lim 𝑥𝑥 = 𝑦𝑥 𝑦)
4645adantr 482 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = 𝑦𝑥 𝑦)
47 vex 3479 . . . . . . . . . 10 𝑥 ∈ V
48 oelim 8534 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4947, 48mpanlr1 705 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5049anasss 468 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5150an12s 648 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
5246, 51sseq12d 4016 . . . . . 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 414 . . . 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 7854 . 2 (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2o) → 𝐵 ⊆ (𝐴o 𝐵)))
5756impcom 409 1 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cdif 3946  wss 3949  c0 4323   cuni 4909   ciun 4998  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367  (class class class)co 7409  2oc2o 8460  o coe 8465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-omul 8471  df-oexp 8472
This theorem is referenced by:  oeeulem  8601  cnfcom3clem  9700  oege2  42105  nnoeomeqom  42110
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