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Theorem oelim2 7562
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 5705 . . . . . 6 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 0ellim 5704 . . . . . . 7 (Lim 𝐵 → ∅ ∈ 𝐵)
32adantl 481 . . . . . 6 ((𝐵𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
4 oe0m1 7488 . . . . . . 7 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
54biimpa 500 . . . . . 6 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
61, 3, 5syl2anc 691 . . . . 5 ((𝐵𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
7 eldif 3550 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ 1𝑜) ↔ (𝑥𝐵 ∧ ¬ 𝑥 ∈ 1𝑜))
8 limord 5701 . . . . . . . . . . . 12 (Lim 𝐵 → Ord 𝐵)
9 ordelon 5664 . . . . . . . . . . . 12 ((Ord 𝐵𝑥𝐵) → 𝑥 ∈ On)
108, 9sylan 487 . . . . . . . . . . 11 ((Lim 𝐵𝑥𝐵) → 𝑥 ∈ On)
11 on0eln0 5697 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
12 el1o 7466 . . . . . . . . . . . . . 14 (𝑥 ∈ 1𝑜𝑥 = ∅)
1312necon3bbii 2829 . . . . . . . . . . . . 13 𝑥 ∈ 1𝑜𝑥 ≠ ∅)
1411, 13syl6bbr 277 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ ¬ 𝑥 ∈ 1𝑜))
15 oe0m1 7488 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
1615biimpd 218 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 → (∅ ↑𝑜 𝑥) = ∅))
1714, 16sylbird 249 . . . . . . . . . . 11 (𝑥 ∈ On → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
1810, 17syl 17 . . . . . . . . . 10 ((Lim 𝐵𝑥𝐵) → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
1918impr 647 . . . . . . . . 9 ((Lim 𝐵 ∧ (𝑥𝐵 ∧ ¬ 𝑥 ∈ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
207, 19sylan2b 491 . . . . . . . 8 ((Lim 𝐵𝑥 ∈ (𝐵 ∖ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
2120iuneq2dv 4478 . . . . . . 7 (Lim 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = 𝑥 ∈ (𝐵 ∖ 1𝑜)∅)
22 df-1o 7447 . . . . . . . . . . 11 1𝑜 = suc ∅
23 limsuc 6941 . . . . . . . . . . . 12 (Lim 𝐵 → (∅ ∈ 𝐵 ↔ suc ∅ ∈ 𝐵))
242, 23mpbid 221 . . . . . . . . . . 11 (Lim 𝐵 → suc ∅ ∈ 𝐵)
2522, 24syl5eqel 2692 . . . . . . . . . 10 (Lim 𝐵 → 1𝑜𝐵)
26 1on 7454 . . . . . . . . . . 11 1𝑜 ∈ On
2726onirri 5751 . . . . . . . . . 10 ¬ 1𝑜 ∈ 1𝑜
2825, 27jctir 559 . . . . . . . . 9 (Lim 𝐵 → (1𝑜𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
29 eldif 3550 . . . . . . . . 9 (1𝑜 ∈ (𝐵 ∖ 1𝑜) ↔ (1𝑜𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
3028, 29sylibr 223 . . . . . . . 8 (Lim 𝐵 → 1𝑜 ∈ (𝐵 ∖ 1𝑜))
31 ne0i 3880 . . . . . . . 8 (1𝑜 ∈ (𝐵 ∖ 1𝑜) → (𝐵 ∖ 1𝑜) ≠ ∅)
32 iunconst 4465 . . . . . . . 8 ((𝐵 ∖ 1𝑜) ≠ ∅ → 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
3330, 31, 323syl 18 . . . . . . 7 (Lim 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
3421, 33eqtrd 2644 . . . . . 6 (Lim 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
3534adantl 481 . . . . 5 ((𝐵𝐶 ∧ Lim 𝐵) → 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
366, 35eqtr4d 2647 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
37 oveq1 6556 . . . . 5 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
38 oveq1 6556 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 𝑥) = (∅ ↑𝑜 𝑥))
3938iuneq2d 4483 . . . . 5 (𝐴 = ∅ → 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
4037, 39eqeq12d 2625 . . . 4 (𝐴 = ∅ → ((𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) ↔ (∅ ↑𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥)))
4136, 40syl5ibr 235 . . 3 (𝐴 = ∅ → ((𝐵𝐶 ∧ Lim 𝐵) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥)))
4241impcom 445 . 2 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
43 oelim 7501 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = 𝑦𝐵 (𝐴𝑜 𝑦))
44 limsuc 6941 . . . . . . . . . . . . 13 (Lim 𝐵 → (𝑦𝐵 ↔ suc 𝑦𝐵))
4544biimpa 500 . . . . . . . . . . . 12 ((Lim 𝐵𝑦𝐵) → suc 𝑦𝐵)
46 nsuceq0 5722 . . . . . . . . . . . . 13 suc 𝑦 ≠ ∅
4746a1i 11 . . . . . . . . . . . 12 ((Lim 𝐵𝑦𝐵) → suc 𝑦 ≠ ∅)
48 dif1o 7467 . . . . . . . . . . . 12 (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ↔ (suc 𝑦𝐵 ∧ suc 𝑦 ≠ ∅))
4945, 47, 48sylanbrc 695 . . . . . . . . . . 11 ((Lim 𝐵𝑦𝐵) → suc 𝑦 ∈ (𝐵 ∖ 1𝑜))
5049ex 449 . . . . . . . . . 10 (Lim 𝐵 → (𝑦𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
5150ad2antlr 759 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
52 sssucid 5719 . . . . . . . . . . 11 𝑦 ⊆ suc 𝑦
53 ordelon 5664 . . . . . . . . . . . . . . . . 17 ((Ord 𝐵𝑦𝐵) → 𝑦 ∈ On)
548, 53sylan 487 . . . . . . . . . . . . . . . 16 ((Lim 𝐵𝑦𝐵) → 𝑦 ∈ On)
55 suceloni 6905 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → suc 𝑦 ∈ On)
5654, 55jccir 560 . . . . . . . . . . . . . . 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 1257 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ On ∧ suc 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
5958ancoms 468 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ (𝑦 ∈ On ∧ suc 𝑦 ∈ On)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
6056, 59sylan2 490 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (Lim 𝐵𝑦𝐵)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
6160anassrs 678 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
62 oewordi 7558 . . . . . . . . . . . . 13 (((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6361, 62sylan 487 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6463an32s 842 . . . . . . . . . . 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 449 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6751, 66jcad 554 . . . . . . . 8 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ∧ (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦))))
68 oveq2 6557 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
6968sseq2d 3596 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥) ↔ (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
7069rspcev 3282 . . . . . . . 8 ((suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ∧ (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)) → ∃𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥))
7167, 70syl6 34 . . . . . . 7 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → ∃𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥)))
7271ralrimiv 2948 . . . . . 6 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∀𝑦𝐵𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥))
73 iunss2 4501 . . . . . 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 3699 . . . . . . . 8 (𝐵 ∖ 1𝑜) ⊆ 𝐵
76 iunss1 4468 . . . . . . . 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 6557 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
7978cbviunv 4495 . . . . . . 7 𝑥𝐵 (𝐴𝑜 𝑥) = 𝑦𝐵 (𝐴𝑜 𝑦)
8077, 79sseqtri 3600 . . . . . 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 3585 . . . 4 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → 𝑦𝐵 (𝐴𝑜 𝑦) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
8382adantlrl 752 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝑦𝐵 (𝐴𝑜 𝑦) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
8443, 83eqtrd 2644 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
8542, 84oe0lem 7480 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 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  cdif 3537  wss 3540  c0 3874   ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1𝑜c1o 7440  𝑜 coe 7446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453
This theorem is referenced by:  oelimcl  7567  oaabs2  7612  omabs  7614
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