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Theorem oalimcl 7873
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 6000 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 oacl 7848 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
3 eloni 5946 . . . 4 ((𝐴 +𝑜 𝐵) ∈ On → Ord (𝐴 +𝑜 𝐵))
42, 3syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +𝑜 𝐵))
51, 4sylan2 582 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +𝑜 𝐵))
6 0ellim 5999 . . . . . 6 (Lim 𝐵 → ∅ ∈ 𝐵)
7 n0i 4121 . . . . . 6 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
86, 7syl 17 . . . . 5 (Lim 𝐵 → ¬ 𝐵 = ∅)
98ad2antll 711 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10 oa00 7872 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11 simpr 473 . . . . . . 7 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐵 = ∅)
1210, 11syl6bi 244 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ → 𝐵 = ∅))
1312con3d 149 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 = ∅ → ¬ (𝐴 +𝑜 𝐵) = ∅))
141, 13sylan2 582 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ 𝐵 = ∅ → ¬ (𝐴 +𝑜 𝐵) = ∅))
159, 14mpd 15 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = ∅)
16 vex 3394 . . . . . . . . . . 11 𝑦 ∈ V
1716sucid 6016 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
18 oalim 7845 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +𝑜 𝐵) = 𝑥𝐵 (𝐴 +𝑜 𝑥))
19 eqeq1 2810 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 +𝑜 𝐵) = 𝑥𝐵 (𝐴 +𝑜 𝑥) ↔ suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥)))
2018, 19syl5ib 235 . . . . . . . . . . 11 ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥)))
2120imp 395 . . . . . . . . . 10 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥))
2217, 21syl5eleq 2891 . . . . . . . . 9 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → 𝑦 𝑥𝐵 (𝐴 +𝑜 𝑥))
23 eliun 4716 . . . . . . . . 9 (𝑦 𝑥𝐵 (𝐴 +𝑜 𝑥) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥))
2422, 23sylib 209 . . . . . . . 8 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → ∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥))
25 onelon 5961 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
261, 25sylan 571 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
27 onnbtwn 6028 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
28 imnan 388 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
2927, 28sylibr 225 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥))
3029com12 32 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3130adantl 469 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3226, 31mpd 15 . . . . . . . . . . . . . 14 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
3332ad2antrl 710 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34 oacl 7848 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ On)
35 eloni 5946 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 +𝑜 𝑥) ∈ On → Ord (𝐴 +𝑜 𝑥))
36 ordsucelsuc 7248 . . . . . . . . . . . . . . . . . . . . . 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 7837 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
3938eleq2d 2871 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +𝑜 𝑥)))
4037, 39bitr4d 273 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
4126, 40sylan2 582 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
42 eleq1 2873 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
4342bicomd 214 . . . . . . . . . . . . . . . . . . 19 ((𝐴 +𝑜 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥) ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
4441, 43sylan9bbr 502 . . . . . . . . . . . . . . . . . 18 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
451adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝐵 ∈ On)
46 sucelon 7243 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
4726, 46sylib 209 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → suc 𝑥 ∈ On)
4845, 47jca 503 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49 oaord 7860 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
50493expa 1140 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 ∈ On ∧ suc 𝑥 ∈ On) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5148, 50sylan 571 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5251ancoms 448 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5352adantl 469 . . . . . . . . . . . . . . . . . 18 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5444, 53bitr4d 273 . . . . . . . . . . . . . . . . 17 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐵 ∈ suc 𝑥))
5554biimpd 220 . . . . . . . . . . . . . . . 16 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → 𝐵 ∈ suc 𝑥))
5655exp32 409 . . . . . . . . . . . . . . 15 ((𝐴 +𝑜 𝐵) = suc 𝑦 → (𝐴 ∈ On → (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → 𝐵 ∈ suc 𝑥))))
5756com4l 92 . . . . . . . . . . . . . 14 (𝐴 ∈ On → (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ((𝐴 +𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))))
5857imp32 407 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ((𝐴 +𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))
5933, 58mtod 189 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
6059exp44 426 . . . . . . . . . . 11 (𝐴 ∈ On → ((𝐵𝐶 ∧ Lim 𝐵) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))))
6160imp 395 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)))
6261rexlimdv 3218 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
6362adantl 469 . . . . . . . 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 400 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ((𝐴 +𝑜 𝐵) = suc 𝑦 → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
6665pm2.01d 181 . . . . 5 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
6766adantr 468 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ On) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
6867nrexdv 3188 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)
69 ioran 997 . . 3 (¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦) ↔ (¬ (𝐴 +𝑜 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
7015, 68, 69sylanbrc 574 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
71 dflim3 7273 . 2 (Lim (𝐴 +𝑜 𝐵) ↔ (Ord (𝐴 +𝑜 𝐵) ∧ ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)))
725, 70, 71sylanbrc 574 1 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2156  wrex 3097  c0 4116   ciun 4712  Ord word 5935  Oncon0 5936  Lim wlim 5937  suc csuc 5938  (class class class)co 6870   +𝑜 coa 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 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
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 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-oadd 7796
This theorem is referenced by:  oaass  7874  odi  7892  wunex3  9844
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