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Theorem oalimcl 7760
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 5901 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 oacl 7735 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
3 eloni 5846 . . . 4 ((𝐴 +𝑜 𝐵) ∈ On → Ord (𝐴 +𝑜 𝐵))
42, 3syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +𝑜 𝐵))
51, 4sylan2 492 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +𝑜 𝐵))
6 0ellim 5900 . . . . . 6 (Lim 𝐵 → ∅ ∈ 𝐵)
7 n0i 4028 . . . . . 6 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
86, 7syl 17 . . . . 5 (Lim 𝐵 → ¬ 𝐵 = ∅)
98ad2antll 767 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10 oa00 7759 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11 simpr 479 . . . . . . 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 492 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ 𝐵 = ∅ → ¬ (𝐴 +𝑜 𝐵) = ∅))
159, 14mpd 15 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = ∅)
16 vex 3307 . . . . . . . . . . 11 𝑦 ∈ V
1716sucid 5917 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
18 oalim 7732 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +𝑜 𝐵) = 𝑥𝐵 (𝐴 +𝑜 𝑥))
19 eqeq1 2728 . . . . . . . . . . . 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 444 . . . . . . . . . 10 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → suc 𝑦 = 𝑥𝐵 (𝐴 +𝑜 𝑥))
2217, 21syl5eleq 2809 . . . . . . . . 9 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → 𝑦 𝑥𝐵 (𝐴 +𝑜 𝑥))
23 eliun 4632 . . . . . . . . 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 5861 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
261, 25sylan 489 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
27 onnbtwn 5931 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
28 imnan 437 . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3226, 31mpd 15 . . . . . . . . . . . . . 14 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
3332ad2antrl 766 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34 oacl 7735 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ On)
35 eloni 5846 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 +𝑜 𝑥) ∈ On → Ord (𝐴 +𝑜 𝑥))
36 ordsucelsuc 7139 . . . . . . . . . . . . . . . . . . . . . 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 7724 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
3938eleq2d 2789 . . . . . . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
42 eleq1 2791 . . . . . . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . . . . . 18 (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
451adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝐵 ∈ On)
46 sucelon 7134 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
4726, 46sylib 208 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → suc 𝑥 ∈ On)
4845, 47jca 555 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49 oaord 7747 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
50493expa 1111 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 ∈ On ∧ suc 𝑥 ∈ On) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5148, 50sylan 489 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5251ancoms 468 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
5352adantl 473 . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . 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 642 . . . . . . . . . . 11 (𝐴 ∈ On → ((𝐵𝐶 ∧ Lim 𝐵) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))))
6160imp 444 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)))
6261rexlimdv 3132 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑥𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
6362adantl 473 . . . . . . . 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 450 . . . . . 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 472 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ On) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
6867nrexdv 3103 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)
69 ioran 512 . . 3 (¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦) ↔ (¬ (𝐴 +𝑜 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
7015, 68, 69sylanbrc 701 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
71 dflim3 7164 . 2 (Lim (𝐴 +𝑜 𝐵) ↔ (Ord (𝐴 +𝑜 𝐵) ∧ ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)))
725, 70, 71sylanbrc 701 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 382  wa 383   = wceq 1596  wcel 2103  wrex 3015  c0 4023   ciun 4628  Ord word 5835  Oncon0 5836  Lim wlim 5837  suc csuc 5838  (class class class)co 6765   +𝑜 coa 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-oadd 7684
This theorem is referenced by:  oaass  7761  odi  7779  wunex3  9676
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