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Theorem oelim2 7906
Description: Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)
Assertion
Ref Expression
oelim2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem oelim2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 limelon 5995 . . . . . 6 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 0ellim 5994 . . . . . . 7 (Lim 𝐵 → ∅ ∈ 𝐵)
32adantl 469 . . . . . 6 ((𝐵𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
4 oe0m1 7832 . . . . . . 7 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
54biimpa 464 . . . . . 6 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
61, 3, 5syl2anc 575 . . . . 5 ((𝐵𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
7 eldif 3773 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ 1𝑜) ↔ (𝑥𝐵 ∧ ¬ 𝑥 ∈ 1𝑜))
8 limord 5991 . . . . . . . . . . . 12 (Lim 𝐵 → Ord 𝐵)
9 ordelon 5954 . . . . . . . . . . . 12 ((Ord 𝐵𝑥𝐵) → 𝑥 ∈ On)
108, 9sylan 571 . . . . . . . . . . 11 ((Lim 𝐵𝑥𝐵) → 𝑥 ∈ On)
11 on0eln0 5987 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
12 el1o 7810 . . . . . . . . . . . . . 14 (𝑥 ∈ 1𝑜𝑥 = ∅)
1312necon3bbii 3021 . . . . . . . . . . . . 13 𝑥 ∈ 1𝑜𝑥 ≠ ∅)
1411, 13syl6bbr 280 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ ¬ 𝑥 ∈ 1𝑜))
15 oe0m1 7832 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
1615biimpd 220 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 → (∅ ↑𝑜 𝑥) = ∅))
1714, 16sylbird 251 . . . . . . . . . . 11 (𝑥 ∈ On → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
1810, 17syl 17 . . . . . . . . . 10 ((Lim 𝐵𝑥𝐵) → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
1918impr 444 . . . . . . . . 9 ((Lim 𝐵 ∧ (𝑥𝐵 ∧ ¬ 𝑥 ∈ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
207, 19sylan2b 583 . . . . . . . 8 ((Lim 𝐵𝑥 ∈ (𝐵 ∖ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
2120iuneq2dv 4727 . . . . . . 7 (Lim 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = 𝑥 ∈ (𝐵 ∖ 1𝑜)∅)
22 df-1o 7790 . . . . . . . . . . 11 1𝑜 = suc ∅
23 limsuc 7273 . . . . . . . . . . . 12 (Lim 𝐵 → (∅ ∈ 𝐵 ↔ suc ∅ ∈ 𝐵))
242, 23mpbid 223 . . . . . . . . . . 11 (Lim 𝐵 → suc ∅ ∈ 𝐵)
2522, 24syl5eqel 2885 . . . . . . . . . 10 (Lim 𝐵 → 1𝑜𝐵)
26 1on 7797 . . . . . . . . . . 11 1𝑜 ∈ On
2726onirri 6041 . . . . . . . . . 10 ¬ 1𝑜 ∈ 1𝑜
2825, 27jctir 512 . . . . . . . . 9 (Lim 𝐵 → (1𝑜𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
29 eldif 3773 . . . . . . . . 9 (1𝑜 ∈ (𝐵 ∖ 1𝑜) ↔ (1𝑜𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
3028, 29sylibr 225 . . . . . . . 8 (Lim 𝐵 → 1𝑜 ∈ (𝐵 ∖ 1𝑜))
31 ne0i 4116 . . . . . . . 8 (1𝑜 ∈ (𝐵 ∖ 1𝑜) → (𝐵 ∖ 1𝑜) ≠ ∅)
32 iunconst 4714 . . . . . . . 8 ((𝐵 ∖ 1𝑜) ≠ ∅ → 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
3330, 31, 323syl 18 . . . . . . 7 (Lim 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
3421, 33eqtrd 2836 . . . . . 6 (Lim 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
3534adantl 469 . . . . 5 ((𝐵𝐶 ∧ Lim 𝐵) → 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
366, 35eqtr4d 2839 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
37 oveq1 6875 . . . . 5 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
38 oveq1 6875 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 𝑥) = (∅ ↑𝑜 𝑥))
3938iuneq2d 4732 . . . . 5 (𝐴 = ∅ → 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
4037, 39eqeq12d 2817 . . . 4 (𝐴 = ∅ → ((𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) ↔ (∅ ↑𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥)))
4136, 40syl5ibr 237 . . 3 (𝐴 = ∅ → ((𝐵𝐶 ∧ Lim 𝐵) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥)))
4241impcom 396 . 2 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
43 oelim 7845 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = 𝑦𝐵 (𝐴𝑜 𝑦))
44 limsuc 7273 . . . . . . . . . . . . 13 (Lim 𝐵 → (𝑦𝐵 ↔ suc 𝑦𝐵))
4544biimpa 464 . . . . . . . . . . . 12 ((Lim 𝐵𝑦𝐵) → suc 𝑦𝐵)
46 nsuceq0 6012 . . . . . . . . . . . . 13 suc 𝑦 ≠ ∅
4746a1i 11 . . . . . . . . . . . 12 ((Lim 𝐵𝑦𝐵) → suc 𝑦 ≠ ∅)
48 dif1o 7811 . . . . . . . . . . . 12 (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ↔ (suc 𝑦𝐵 ∧ suc 𝑦 ≠ ∅))
4945, 47, 48sylanbrc 574 . . . . . . . . . . 11 ((Lim 𝐵𝑦𝐵) → suc 𝑦 ∈ (𝐵 ∖ 1𝑜))
5049ex 399 . . . . . . . . . 10 (Lim 𝐵 → (𝑦𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
5150ad2antlr 709 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
52 sssucid 6009 . . . . . . . . . . 11 𝑦 ⊆ suc 𝑦
53 ordelon 5954 . . . . . . . . . . . . . . . . 17 ((Ord 𝐵𝑦𝐵) → 𝑦 ∈ On)
548, 53sylan 571 . . . . . . . . . . . . . . . 16 ((Lim 𝐵𝑦𝐵) → 𝑦 ∈ On)
55 suceloni 7237 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → suc 𝑦 ∈ On)
5654, 55jccir 513 . . . . . . . . . . . . . . 15 ((Lim 𝐵𝑦𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On))
57 id 22 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
58573expa 1140 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ On ∧ suc 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
5958ancoms 448 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ (𝑦 ∈ On ∧ suc 𝑦 ∈ On)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
6056, 59sylan2 582 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (Lim 𝐵𝑦𝐵)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
6160anassrs 455 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
62 oewordi 7902 . . . . . . . . . . . . 13 (((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6361, 62sylan 571 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6463an32s 634 . . . . . . . . . . 11 ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) ∧ 𝑦𝐵) → (𝑦 ⊆ suc 𝑦 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6552, 64mpi 20 . . . . . . . . . 10 ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) ∧ 𝑦𝐵) → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦))
6665ex 399 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6751, 66jcad 504 . . . . . . . 8 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ∧ (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦))))
68 oveq2 6876 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
6968sseq2d 3824 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥) ↔ (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
7069rspcev 3498 . . . . . . . 8 ((suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ∧ (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)) → ∃𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥))
7167, 70syl6 35 . . . . . . 7 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → ∃𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥)))
7271ralrimiv 3149 . . . . . 6 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∀𝑦𝐵𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥))
73 iunss2 4750 . . . . . 6 (∀𝑦𝐵𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥) → 𝑦𝐵 (𝐴𝑜 𝑦) ⊆ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
7472, 73syl 17 . . . . 5 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → 𝑦𝐵 (𝐴𝑜 𝑦) ⊆ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
75 difss 3930 . . . . . . . 8 (𝐵 ∖ 1𝑜) ⊆ 𝐵
76 iunss1 4717 . . . . . . . 8 ((𝐵 ∖ 1𝑜) ⊆ 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) ⊆ 𝑥𝐵 (𝐴𝑜 𝑥))
7775, 76ax-mp 5 . . . . . . 7 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) ⊆ 𝑥𝐵 (𝐴𝑜 𝑥)
78 oveq2 6876 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
7978cbviunv 4744 . . . . . . 7 𝑥𝐵 (𝐴𝑜 𝑥) = 𝑦𝐵 (𝐴𝑜 𝑦)
8077, 79sseqtri 3828 . . . . . 6 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) ⊆ 𝑦𝐵 (𝐴𝑜 𝑦)
8180a1i 11 . . . . 5 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) ⊆ 𝑦𝐵 (𝐴𝑜 𝑦))
8274, 81eqssd 3809 . . . 4 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → 𝑦𝐵 (𝐴𝑜 𝑦) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
8382adantlrl 702 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝑦𝐵 (𝐴𝑜 𝑦) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
8443, 83eqtrd 2836 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
8542, 84oe0lem 7824 1 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2155  wne 2974  wral 3092  wrex 3093  cdif 3760  wss 3763  c0 4110   ciun 4705  Ord word 5929  Oncon0 5930  Lim wlim 5931  suc csuc 5932  (class class class)co 6868  1𝑜c1o 7783  𝑜 coe 7789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-2o 7791  df-oadd 7794  df-omul 7795  df-oexp 7796
This theorem is referenced by:  oelimcl  7911  oaabs2  7956  omabs  7958
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