MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oalimcl Structured version   Visualization version   GIF version

Theorem oalimcl 8512
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 6386 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 oacl 8486 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
3 eloni 6332 . . . 4 ((𝐴 +o 𝐵) ∈ On → Ord (𝐴 +o 𝐵))
42, 3syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +o 𝐵))
51, 4sylan2 594 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +o 𝐵))
6 0ellim 6385 . . . . . 6 (Lim 𝐵 → ∅ ∈ 𝐵)
7 n0i 4298 . . . . . 6 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
86, 7syl 17 . . . . 5 (Lim 𝐵 → ¬ 𝐵 = ∅)
98ad2antll 728 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10 oa00 8511 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11 simpr 486 . . . . . . 7 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐵 = ∅)
1210, 11syl6bi 253 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ → 𝐵 = ∅))
1312con3d 152 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 = ∅ → ¬ (𝐴 +o 𝐵) = ∅))
141, 13sylan2 594 . . . 4 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ 𝐵 = ∅ → ¬ (𝐴 +o 𝐵) = ∅))
159, 14mpd 15 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +o 𝐵) = ∅)
16 vex 3452 . . . . . . . . . . 11 𝑦 ∈ V
1716sucid 6404 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
18 oalim 8483 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥))
19 eqeq1 2741 . . . . . . . . . . . 12 ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 +o 𝐵) = 𝑥𝐵 (𝐴 +o 𝑥) ↔ suc 𝑦 = 𝑥𝐵 (𝐴 +o 𝑥)))
2018, 19imbitrid 243 . . . . . . . . . . 11 ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → suc 𝑦 = 𝑥𝐵 (𝐴 +o 𝑥)))
2120imp 408 . . . . . . . . . 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 4963 . . . . . . . . 9 (𝑦 𝑥𝐵 (𝐴 +o 𝑥) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴 +o 𝑥))
2422, 23sylib 217 . . . . . . . 8 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵))) → ∃𝑥𝐵 𝑦 ∈ (𝐴 +o 𝑥))
25 onelon 6347 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
261, 25sylan 581 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
27 onnbtwn 6416 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
28 imnan 401 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
2927, 28sylibr 233 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥))
3029com12 32 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3130adantl 483 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3226, 31mpd 15 . . . . . . . . . . . . . 14 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
3332ad2antrl 727 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34 oacl 8486 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ On)
35 eloni 6332 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 +o 𝑥) ∈ On → Ord (𝐴 +o 𝑥))
36 ordsucelsuc 7762 . . . . . . . . . . . . . . . . . . . . . 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 8475 . . . . . . . . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . 19 ((𝐴 +o 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 +o suc 𝑥) ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
4441, 43sylan9bbr 512 . . . . . . . . . . . . . . . . . 18 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
451adantr 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝐵 ∈ On)
46 onsucb 7757 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
4726, 46sylib 217 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → suc 𝑥 ∈ On)
4845, 47jca 513 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49 oaord 8499 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
50493expa 1119 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 ∈ On ∧ suc 𝑥 ∈ On) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
5148, 50sylan 581 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
5251ancoms 460 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
5352adantl 483 . . . . . . . . . . . . . . . . . 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 228 . . . . . . . . . . . . . . . 16 (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) → 𝐵 ∈ suc 𝑥))
5655exp32 422 . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ((𝐴 +o 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))
5933, 58mtod 197 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
6059exp44 439 . . . . . . . . . . 11 (𝐴 ∈ On → ((𝐵𝐶 ∧ Lim 𝐵) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦))))
6160imp 408 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦)))
6261rexlimdv 3151 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑥𝐵 𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦))
6362adantl 483 . . . . . . . 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 415 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ((𝐴 +o 𝐵) = suc 𝑦 → ¬ (𝐴 +o 𝐵) = suc 𝑦))
6665pm2.01d 189 . . . . 5 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
6766adantr 482 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ On) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
6867nrexdv 3147 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦)
69 ioran 983 . . 3 (¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦) ↔ (¬ (𝐴 +o 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦))
7015, 68, 69sylanbrc 584 . 2 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦))
71 dflim3 7788 . 2 (Lim (𝐴 +o 𝐵) ↔ (Ord (𝐴 +o 𝐵) ∧ ¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦)))
725, 70, 71sylanbrc 584 1 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wrex 3074  c0 4287   ciun 4959  Ord word 6321  Oncon0 6322  Lim wlim 6323  suc csuc 6324  (class class class)co 7362   +o coa 8414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-oadd 8421
This theorem is referenced by:  oaass  8513  odi  8531  wunex3  10684  omlimcl2  41605  oalim2cl  41653
  Copyright terms: Public domain W3C validator