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Theorem oelim2 7535
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 5687 . . . . . 6 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 0ellim 5686 . . . . . . 7 (Lim 𝐵 → ∅ ∈ 𝐵)
32adantl 480 . . . . . 6 ((𝐵𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
4 oe0m1 7461 . . . . . . 7 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
54biimpa 499 . . . . . 6 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
61, 3, 5syl2anc 690 . . . . 5 ((𝐵𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
7 eldif 3545 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ 1𝑜) ↔ (𝑥𝐵 ∧ ¬ 𝑥 ∈ 1𝑜))
8 limord 5683 . . . . . . . . . . . 12 (Lim 𝐵 → Ord 𝐵)
9 ordelon 5646 . . . . . . . . . . . 12 ((Ord 𝐵𝑥𝐵) → 𝑥 ∈ On)
108, 9sylan 486 . . . . . . . . . . 11 ((Lim 𝐵𝑥𝐵) → 𝑥 ∈ On)
11 on0eln0 5679 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
12 el1o 7439 . . . . . . . . . . . . . 14 (𝑥 ∈ 1𝑜𝑥 = ∅)
1312necon3bbii 2824 . . . . . . . . . . . . 13 𝑥 ∈ 1𝑜𝑥 ≠ ∅)
1411, 13syl6bbr 276 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ ¬ 𝑥 ∈ 1𝑜))
15 oe0m1 7461 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
1615biimpd 217 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 → (∅ ↑𝑜 𝑥) = ∅))
1714, 16sylbird 248 . . . . . . . . . . 11 (𝑥 ∈ On → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
1810, 17syl 17 . . . . . . . . . 10 ((Lim 𝐵𝑥𝐵) → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
1918impr 646 . . . . . . . . 9 ((Lim 𝐵 ∧ (𝑥𝐵 ∧ ¬ 𝑥 ∈ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
207, 19sylan2b 490 . . . . . . . 8 ((Lim 𝐵𝑥 ∈ (𝐵 ∖ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
2120iuneq2dv 4468 . . . . . . 7 (Lim 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = 𝑥 ∈ (𝐵 ∖ 1𝑜)∅)
22 df-1o 7420 . . . . . . . . . . 11 1𝑜 = suc ∅
23 limsuc 6914 . . . . . . . . . . . 12 (Lim 𝐵 → (∅ ∈ 𝐵 ↔ suc ∅ ∈ 𝐵))
242, 23mpbid 220 . . . . . . . . . . 11 (Lim 𝐵 → suc ∅ ∈ 𝐵)
2522, 24syl5eqel 2687 . . . . . . . . . 10 (Lim 𝐵 → 1𝑜𝐵)
26 1on 7427 . . . . . . . . . . 11 1𝑜 ∈ On
2726onirri 5733 . . . . . . . . . 10 ¬ 1𝑜 ∈ 1𝑜
2825, 27jctir 558 . . . . . . . . 9 (Lim 𝐵 → (1𝑜𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
29 eldif 3545 . . . . . . . . 9 (1𝑜 ∈ (𝐵 ∖ 1𝑜) ↔ (1𝑜𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
3028, 29sylibr 222 . . . . . . . 8 (Lim 𝐵 → 1𝑜 ∈ (𝐵 ∖ 1𝑜))
31 ne0i 3875 . . . . . . . 8 (1𝑜 ∈ (𝐵 ∖ 1𝑜) → (𝐵 ∖ 1𝑜) ≠ ∅)
32 iunconst 4455 . . . . . . . 8 ((𝐵 ∖ 1𝑜) ≠ ∅ → 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
3330, 31, 323syl 18 . . . . . . 7 (Lim 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
3421, 33eqtrd 2639 . . . . . 6 (Lim 𝐵 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
3534adantl 480 . . . . 5 ((𝐵𝐶 ∧ Lim 𝐵) → 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
366, 35eqtr4d 2642 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
37 oveq1 6530 . . . . 5 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
38 oveq1 6530 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 𝑥) = (∅ ↑𝑜 𝑥))
3938iuneq2d 4473 . . . . 5 (𝐴 = ∅ → 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
4037, 39eqeq12d 2620 . . . 4 (𝐴 = ∅ → ((𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥) ↔ (∅ ↑𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥)))
4136, 40syl5ibr 234 . . 3 (𝐴 = ∅ → ((𝐵𝐶 ∧ Lim 𝐵) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥)))
4241impcom 444 . 2 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
43 oelim 7474 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = 𝑦𝐵 (𝐴𝑜 𝑦))
44 limsuc 6914 . . . . . . . . . . . . 13 (Lim 𝐵 → (𝑦𝐵 ↔ suc 𝑦𝐵))
4544biimpa 499 . . . . . . . . . . . 12 ((Lim 𝐵𝑦𝐵) → suc 𝑦𝐵)
46 nsuceq0 5704 . . . . . . . . . . . . 13 suc 𝑦 ≠ ∅
4746a1i 11 . . . . . . . . . . . 12 ((Lim 𝐵𝑦𝐵) → suc 𝑦 ≠ ∅)
48 dif1o 7440 . . . . . . . . . . . 12 (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ↔ (suc 𝑦𝐵 ∧ suc 𝑦 ≠ ∅))
4945, 47, 48sylanbrc 694 . . . . . . . . . . 11 ((Lim 𝐵𝑦𝐵) → suc 𝑦 ∈ (𝐵 ∖ 1𝑜))
5049ex 448 . . . . . . . . . 10 (Lim 𝐵 → (𝑦𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
5150ad2antlr 758 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
52 sssucid 5701 . . . . . . . . . . 11 𝑦 ⊆ suc 𝑦
53 ordelon 5646 . . . . . . . . . . . . . . . . 17 ((Ord 𝐵𝑦𝐵) → 𝑦 ∈ On)
548, 53sylan 486 . . . . . . . . . . . . . . . 16 ((Lim 𝐵𝑦𝐵) → 𝑦 ∈ On)
55 suceloni 6878 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → suc 𝑦 ∈ On)
5654, 55jccir 559 . . . . . . . . . . . . . . 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 1256 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ On ∧ suc 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
5958ancoms 467 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ (𝑦 ∈ On ∧ suc 𝑦 ∈ On)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
6056, 59sylan2 489 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (Lim 𝐵𝑦𝐵)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
6160anassrs 677 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
62 oewordi 7531 . . . . . . . . . . . . 13 (((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6361, 62sylan 486 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6463an32s 841 . . . . . . . . . . 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 448 . . . . . . . . 9 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
6751, 66jcad 553 . . . . . . . 8 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦𝐵 → (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ∧ (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦))))
68 oveq2 6531 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
6968sseq2d 3591 . . . . . . . . 9 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥) ↔ (𝐴𝑜 𝑦) ⊆ (𝐴𝑜 suc 𝑦)))
7069rspcev 3277 . . . . . . . 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 2943 . . . . . 6 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∀𝑦𝐵𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ⊆ (𝐴𝑜 𝑥))
73 iunss2 4491 . . . . . 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 3694 . . . . . . . 8 (𝐵 ∖ 1𝑜) ⊆ 𝐵
76 iunss1 4458 . . . . . . . 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 6531 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
7978cbviunv 4485 . . . . . . 7 𝑥𝐵 (𝐴𝑜 𝑥) = 𝑦𝐵 (𝐴𝑜 𝑦)
8077, 79sseqtri 3595 . . . . . 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 3580 . . . 4 (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → 𝑦𝐵 (𝐴𝑜 𝑦) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
8382adantlrl 751 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝑦𝐵 (𝐴𝑜 𝑦) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
8443, 83eqtrd 2639 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))
8542, 84oe0lem 7453 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 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wral 2891  wrex 2892  cdif 3532  wss 3535  c0 3869   ciun 4445  Ord word 5621  Oncon0 5622  Lim wlim 5623  suc csuc 5624  (class class class)co 6523  1𝑜c1o 7413  𝑜 coe 7419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-omul 7425  df-oexp 7426
This theorem is referenced by:  oelimcl  7540  oaabs2  7585  omabs  7587
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