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Theorem oalimcl 8596
Description: The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. Lemma 3.4 of [Schloeder] p. 7. (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 6449 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 oacl 8571 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
3 eloni 6395 . . . 4 ((𝐴 +o 𝐵) ∈ On → Ord (𝐴 +o 𝐵))
42, 3syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +o 𝐵))
51, 4sylan2 593 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +o 𝐵))
6 0ellim 6448 . . . . . 6 (Lim 𝐵 → ∅ ∈ 𝐵)
7 n0i 4345 . . . . . 6 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
86, 7syl 17 . . . . 5 (Lim 𝐵 → ¬ 𝐵 = ∅)
98ad2antll 729 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10 oa00 8595 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11 simpr 484 . . . . . . 7 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐵 = ∅)
1210, 11biimtrdi 253 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ → 𝐵 = ∅))
1312con3d 152 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 = ∅ → ¬ (𝐴 +o 𝐵) = ∅))
141, 13sylan2 593 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ 𝐵 = ∅ → ¬ (𝐴 +o 𝐵) = ∅))
159, 14mpd 15 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +o 𝐵) = ∅)
16 vex 3481 . . . . . . . . . . 11 𝑦 ∈ V
1716sucid 6467 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
18 oalim 8568 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥))
19 eqeq1 2738 . . . . . . . . . . . 12 ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥) ↔ suc 𝑦 = 𝑥𝐵 (𝐴 +o 𝑥)))
2018, 19imbitrid 244 . . . . . . . . . . 11 ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → suc 𝑦 = 𝑥𝐵 (𝐴 +o 𝑥)))
2120imp 406 . . . . . . . . . 10 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → suc 𝑦 = 𝑥𝐵 (𝐴 +o 𝑥))
2217, 21eleqtrid 2844 . . . . . . . . 9 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → 𝑦 𝑥𝐵 (𝐴 +o 𝑥))
23 eliun 4999 . . . . . . . . 9 (𝑦 𝑥𝐵 (𝐴 +o 𝑥) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴 +o 𝑥))
2422, 23sylib 218 . . . . . . . 8 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → ∃𝑥𝐵 𝑦 ∈ (𝐴 +o 𝑥))
25 onelon 6410 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
261, 25sylan 580 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
27 onnbtwn 6479 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
28 imnan 399 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
2927, 28sylibr 234 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥))
3029com12 32 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3130adantl 481 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3226, 31mpd 15 . . . . . . . . . . . . . 14 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
3332ad2antrl 728 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34 oacl 8571 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ On)
35 eloni 6395 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 +o 𝑥) ∈ On → Ord (𝐴 +o 𝑥))
36 ordsucelsuc 7841 . . . . . . . . . . . . . . . . . . . . . 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 8560 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
3938eleq2d 2824 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (suc 𝑦 ∈ (𝐴 +o suc 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +o 𝑥)))
4037, 39bitr4d 282 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
4126, 40sylan2 593 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
42 eleq1 2826 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
4342bicomd 223 . . . . . . . . . . . . . . . . . . 19 ((𝐴 +o 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 +o suc 𝑥) ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
4441, 43sylan9bbr 510 . . . . . . . . . . . . . . . . . 18 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
451adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝐵 ∈ On)
46 onsucb 7836 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
4726, 46sylib 218 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → suc 𝑥 ∈ On)
4845, 47jca 511 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49 oaord 8583 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
50493expa 1117 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 ∈ On ∧ suc 𝑥 ∈ On) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
5148, 50sylan 580 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
5251ancoms 458 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
5352adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
5444, 53bitr4d 282 . . . . . . . . . . . . . . . . 17 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ 𝐵 ∈ suc 𝑥))
5554biimpd 229 . . . . . . . . . . . . . . . 16 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) → 𝐵 ∈ suc 𝑥))
5655exp32 420 . . . . . . . . . . . . . . 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 418 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ((𝐴 +o 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))
5933, 58mtod 198 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
6059exp44 437 . . . . . . . . . . 11 (𝐴 ∈ On → ((𝐵𝐶 ∧ Lim 𝐵) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦))))
6160imp 406 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦)))
6261rexlimdv 3150 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑥𝐵 𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦))
6362adantl 481 . . . . . . . 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 413 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ((𝐴 +o 𝐵) = suc 𝑦 → ¬ (𝐴 +o 𝐵) = suc 𝑦))
6665pm2.01d 190 . . . . 5 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
6766adantr 480 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ On) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
6867nrexdv 3146 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦)
69 ioran 985 . . 3 (¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦) ↔ (¬ (𝐴 +o 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦))
7015, 68, 69sylanbrc 583 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦))
71 dflim3 7867 . 2 (Lim (𝐴 +o 𝐵) ↔ (Ord (𝐴 +o 𝐵) ∧ ¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦)))
725, 70, 71sylanbrc 583 1 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wrex 3067  c0 4338   ciun 4995  Ord word 6384  Oncon0 6385  Lim wlim 6386  suc csuc 6387  (class class class)co 7430   +o coa 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-oadd 8508
This theorem is referenced by:  oaass  8597  odi  8615  wunex3  10778  omlimcl2  43230  oalim2cl  43278
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