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Theorem oalimcl 7500
Description: The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. (Contributed by NM, 8-Dec-2004.)
Assertion
Ref Expression
oalimcl ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +𝑜 𝐵))

Proof of Theorem oalimcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limelon 5687 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 oacl 7475 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
3 eloni 5632 . . . 4 ((𝐴 +𝑜 𝐵) ∈ On → Ord (𝐴 +𝑜 𝐵))
42, 3syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +𝑜 𝐵))
51, 4sylan2 489 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +𝑜 𝐵))
6 0ellim 5686 . . . . . 6 (Lim 𝐵 → ∅ ∈ 𝐵)
7 n0i 3874 . . . . . 6 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
86, 7syl 17 . . . . 5 (Lim 𝐵 → ¬ 𝐵 = ∅)
98ad2antll 760 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10 oa00 7499 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11 simpr 475 . . . . . . 7 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐵 = ∅)
1210, 11syl6bi 241 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ → 𝐵 = ∅))
1312con3d 146 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 = ∅ → ¬ (𝐴 +𝑜 𝐵) = ∅))
141, 13sylan2 489 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ 𝐵 = ∅ → ¬ (𝐴 +𝑜 𝐵) = ∅))
159, 14mpd 15 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = ∅)
16 vex 3171 . . . . . . . . . . 11 𝑦 ∈ V
1716sucid 5703 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
18 oalim 7472 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +𝑜 𝐵) = 𝑥𝐵 (𝐴 +𝑜 𝑥))
19 eqeq1 2609 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 +𝑜 𝐵) = 𝑥𝐵 (𝐴 +𝑜 𝑥) ↔ suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥)))
2018, 19syl5ib 232 . . . . . . . . . . 11 ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥)))
2120imp 443 . . . . . . . . . 10 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥))
2217, 21syl5eleq 2689 . . . . . . . . 9 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → 𝑦 𝑥𝐵 (𝐴 +𝑜 𝑥))
23 eliun 4450 . . . . . . . . 9 (𝑦 𝑥𝐵 (𝐴 +𝑜 𝑥) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥))
2422, 23sylib 206 . . . . . . . 8 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → ∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥))
25 onelon 5647 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
261, 25sylan 486 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
27 onnbtwn 5717 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
28 imnan 436 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
2927, 28sylibr 222 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥))
3029com12 32 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3130adantl 480 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3226, 31mpd 15 . . . . . . . . . . . . . 14 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
3332ad2antrl 759 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34 oacl 7475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ On)
35 eloni 5632 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 +𝑜 𝑥) ∈ On → Ord (𝐴 +𝑜 𝑥))
36 ordsucelsuc 6887 . . . . . . . . . . . . . . . . . . . . . 22 (Ord (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +𝑜 𝑥)))
3734, 35, 363syl 18 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +𝑜 𝑥)))
38 oasuc 7464 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
3938eleq2d 2668 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +𝑜 𝑥)))
4037, 39bitr4d 269 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
4126, 40sylan2 489 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
42 eleq1 2671 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
4342bicomd 211 . . . . . . . . . . . . . . . . . . 19 ((𝐴 +𝑜 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥) ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
4441, 43sylan9bbr 732 . . . . . . . . . . . . . . . . . 18 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
451adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝐵 ∈ On)
46 sucelon 6882 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
4726, 46sylib 206 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → suc 𝑥 ∈ On)
4845, 47jca 552 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49 oaord 7487 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
50493expa 1256 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 ∈ On ∧ suc 𝑥 ∈ On) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5148, 50sylan 486 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5251ancoms 467 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5352adantl 480 . . . . . . . . . . . . . . . . . 18 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5444, 53bitr4d 269 . . . . . . . . . . . . . . . . 17 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐵 ∈ suc 𝑥))
5554biimpd 217 . . . . . . . . . . . . . . . 16 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → 𝐵 ∈ suc 𝑥))
5655exp32 628 . . . . . . . . . . . . . . 15 ((𝐴 +𝑜 𝐵) = suc 𝑦 → (𝐴 ∈ On → (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → 𝐵 ∈ suc 𝑥))))
5756com4l 89 . . . . . . . . . . . . . 14 (𝐴 ∈ On → (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ((𝐴 +𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))))
5857imp32 447 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ((𝐴 +𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))
5933, 58mtod 187 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
6059exp44 638 . . . . . . . . . . 11 (𝐴 ∈ On → ((𝐵𝐶 ∧ Lim 𝐵) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))))
6160imp 443 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)))
6261rexlimdv 3007 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
6362adantl 480 . . . . . . . 8 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → (∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
6424, 63mpd 15 . . . . . . 7 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
6564expcom 449 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ((𝐴 +𝑜 𝐵) = suc 𝑦 → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
6665pm2.01d 179 . . . . 5 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
6766adantr 479 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ On) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
6867nrexdv 2979 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)
69 ioran 509 . . 3 (¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦) ↔ (¬ (𝐴 +𝑜 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
7015, 68, 69sylanbrc 694 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
71 dflim3 6912 . 2 (Lim (𝐴 +𝑜 𝐵) ↔ (Ord (𝐴 +𝑜 𝐵) ∧ ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)))
725, 70, 71sylanbrc 694 1 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1975  wrex 2892  c0 3869   ciun 4445  Ord word 5621  Oncon0 5622  Lim wlim 5623  suc csuc 5624  (class class class)co 6523   +𝑜 coa 7417
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-oadd 7424
This theorem is referenced by:  oaass  7501  odi  7519  wunex3  9415
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