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Theorem oalimcl 8181
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 6253 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 oacl 8156 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
3 eloni 6200 . . . 4 ((𝐴 +o 𝐵) ∈ On → Ord (𝐴 +o 𝐵))
42, 3syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +o 𝐵))
51, 4sylan2 592 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +o 𝐵))
6 0ellim 6252 . . . . . 6 (Lim 𝐵 → ∅ ∈ 𝐵)
7 n0i 4303 . . . . . 6 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
86, 7syl 17 . . . . 5 (Lim 𝐵 → ¬ 𝐵 = ∅)
98ad2antll 725 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10 oa00 8180 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11 simpr 485 . . . . . . 7 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐵 = ∅)
1210, 11syl6bi 254 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ → 𝐵 = ∅))
1312con3d 155 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 = ∅ → ¬ (𝐴 +o 𝐵) = ∅))
141, 13sylan2 592 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ 𝐵 = ∅ → ¬ (𝐴 +o 𝐵) = ∅))
159, 14mpd 15 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +o 𝐵) = ∅)
16 vex 3503 . . . . . . . . . . 11 𝑦 ∈ V
1716sucid 6269 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
18 oalim 8153 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥))
19 eqeq1 2830 . . . . . . . . . . . 12 ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥) ↔ suc 𝑦 = 𝑥𝐵 (𝐴 +o 𝑥)))
2018, 19syl5ib 245 . . . . . . . . . . 11 ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → suc 𝑦 = 𝑥𝐵 (𝐴 +o 𝑥)))
2120imp 407 . . . . . . . . . 10 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → suc 𝑦 = 𝑥𝐵 (𝐴 +o 𝑥))
2217, 21eleqtrid 2924 . . . . . . . . 9 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → 𝑦 𝑥𝐵 (𝐴 +o 𝑥))
23 eliun 4921 . . . . . . . . 9 (𝑦 𝑥𝐵 (𝐴 +o 𝑥) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴 +o 𝑥))
2422, 23sylib 219 . . . . . . . 8 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → ∃𝑥𝐵 𝑦 ∈ (𝐴 +o 𝑥))
25 onelon 6215 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
261, 25sylan 580 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
27 onnbtwn 6281 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
28 imnan 400 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
2927, 28sylibr 235 . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34 oacl 8156 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ On)
35 eloni 6200 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 +o 𝑥) ∈ On → Ord (𝐴 +o 𝑥))
36 ordsucelsuc 7530 . . . . . . . . . . . . . . . . . . . . . 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 8145 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
3938eleq2d 2903 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (suc 𝑦 ∈ (𝐴 +o suc 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +o 𝑥)))
4037, 39bitr4d 283 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
4126, 40sylan2 592 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
42 eleq1 2905 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
4342bicomd 224 . . . . . . . . . . . . . . . . . . 19 ((𝐴 +o 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 +o suc 𝑥) ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
4441, 43sylan9bbr 511 . . . . . . . . . . . . . . . . . 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 7525 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
4726, 46sylib 219 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → suc 𝑥 ∈ On)
4845, 47jca 512 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49 oaord 8168 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
50493expa 1112 . . . . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . 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 283 . . . . . . . . . . . . . . . . 17 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ 𝐵 ∈ suc 𝑥))
5554biimpd 230 . . . . . . . . . . . . . . . 16 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) → 𝐵 ∈ suc 𝑥))
5655exp32 421 . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ((𝐴 +o 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))
5933, 58mtod 199 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
6059exp44 438 . . . . . . . . . . 11 (𝐴 ∈ On → ((𝐵𝐶 ∧ Lim 𝐵) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦))))
6160imp 407 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦)))
6261rexlimdv 3288 . . . . . . . . 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 414 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ((𝐴 +o 𝐵) = suc 𝑦 → ¬ (𝐴 +o 𝐵) = suc 𝑦))
6665pm2.01d 191 . . . . 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 3275 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦)
69 ioran 979 . . 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 7555 . 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 207  wa 396  wo 843   = wceq 1530  wcel 2107  wrex 3144  c0 4295   ciun 4917  Ord word 6189  Oncon0 6190  Lim wlim 6191  suc csuc 6192  (class class class)co 7150   +o coa 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-oadd 8102
This theorem is referenced by:  oaass  8182  odi  8200  wunex3  10157
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