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Theorem oalimcl 7600
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 5757 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 oacl 7575 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
3 eloni 5702 . . . 4 ((𝐴 +𝑜 𝐵) ∈ On → Ord (𝐴 +𝑜 𝐵))
42, 3syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +𝑜 𝐵))
51, 4sylan2 491 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +𝑜 𝐵))
6 0ellim 5756 . . . . . 6 (Lim 𝐵 → ∅ ∈ 𝐵)
7 n0i 3902 . . . . . 6 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
86, 7syl 17 . . . . 5 (Lim 𝐵 → ¬ 𝐵 = ∅)
98ad2antll 764 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10 oa00 7599 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11 simpr 477 . . . . . . 7 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐵 = ∅)
1210, 11syl6bi 243 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ → 𝐵 = ∅))
1312con3d 148 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 = ∅ → ¬ (𝐴 +𝑜 𝐵) = ∅))
141, 13sylan2 491 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ 𝐵 = ∅ → ¬ (𝐴 +𝑜 𝐵) = ∅))
159, 14mpd 15 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = ∅)
16 vex 3193 . . . . . . . . . . 11 𝑦 ∈ V
1716sucid 5773 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
18 oalim 7572 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +𝑜 𝐵) = 𝑥𝐵 (𝐴 +𝑜 𝑥))
19 eqeq1 2625 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 +𝑜 𝐵) = 𝑥𝐵 (𝐴 +𝑜 𝑥) ↔ suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥)))
2018, 19syl5ib 234 . . . . . . . . . . 11 ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥)))
2120imp 445 . . . . . . . . . 10 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥))
2217, 21syl5eleq 2704 . . . . . . . . 9 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → 𝑦 𝑥𝐵 (𝐴 +𝑜 𝑥))
23 eliun 4497 . . . . . . . . 9 (𝑦 𝑥𝐵 (𝐴 +𝑜 𝑥) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥))
2422, 23sylib 208 . . . . . . . 8 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → ∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥))
25 onelon 5717 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
261, 25sylan 488 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
27 onnbtwn 5787 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
28 imnan 438 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
2927, 28sylibr 224 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥))
3029com12 32 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3130adantl 482 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3226, 31mpd 15 . . . . . . . . . . . . . 14 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
3332ad2antrl 763 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34 oacl 7575 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ On)
35 eloni 5702 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 +𝑜 𝑥) ∈ On → Ord (𝐴 +𝑜 𝑥))
36 ordsucelsuc 6984 . . . . . . . . . . . . . . . . . . . . . 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 7564 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
3938eleq2d 2684 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +𝑜 𝑥)))
4037, 39bitr4d 271 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
4126, 40sylan2 491 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
42 eleq1 2686 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
4342bicomd 213 . . . . . . . . . . . . . . . . . . 19 ((𝐴 +𝑜 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥) ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
4441, 43sylan9bbr 736 . . . . . . . . . . . . . . . . . 18 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
451adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝐵 ∈ On)
46 sucelon 6979 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
4726, 46sylib 208 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → suc 𝑥 ∈ On)
4845, 47jca 554 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49 oaord 7587 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
50493expa 1262 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 ∈ On ∧ suc 𝑥 ∈ On) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5148, 50sylan 488 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5251ancoms 469 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5352adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5444, 53bitr4d 271 . . . . . . . . . . . . . . . . 17 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐵 ∈ suc 𝑥))
5554biimpd 219 . . . . . . . . . . . . . . . 16 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → 𝐵 ∈ suc 𝑥))
5655exp32 630 . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . 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 640 . . . . . . . . . . 11 (𝐴 ∈ On → ((𝐵𝐶 ∧ Lim 𝐵) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))))
6160imp 445 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)))
6261rexlimdv 3025 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
6362adantl 482 . . . . . . . 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 451 . . . . . 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 481 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ On) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
6867nrexdv 2997 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)
69 ioran 511 . . 3 (¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦) ↔ (¬ (𝐴 +𝑜 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
7015, 68, 69sylanbrc 697 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
71 dflim3 7009 . 2 (Lim (𝐴 +𝑜 𝐵) ↔ (Ord (𝐴 +𝑜 𝐵) ∧ ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)))
725, 70, 71sylanbrc 697 1 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wrex 2909  c0 3897   ciun 4492  Ord word 5691  Oncon0 5692  Lim wlim 5693  suc csuc 5694  (class class class)co 6615   +𝑜 coa 7517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524
This theorem is referenced by:  oaass  7601  odi  7619  wunex3  9523
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