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Theorem oelimcl 8213
 Description: The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
oelimcl ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴o 𝐵))

Proof of Theorem oelimcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 4078 . . . 4 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2 limelon 6232 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3 oecl 8149 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
41, 2, 3syl2an 598 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) ∈ On)
5 eloni 6179 . . 3 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
64, 5syl 17 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴o 𝐵))
71adantr 484 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
82adantl 485 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9 dif20el 8117 . . . 4 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
109adantr 484 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11 oen0 8199 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵))
127, 8, 10, 11syl21anc 836 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴o 𝐵))
13 oelim2 8208 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) = 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦))
141, 13sylan 583 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) = 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦))
1514eleq2d 2899 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴o 𝐵) ↔ 𝑥 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦)))
16 eliun 4898 . . . . 5 (𝑥 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴o 𝑦))
17 eldifi 4078 . . . . . . 7 (𝑦 ∈ (𝐵 ∖ 1o) → 𝑦𝐵)
187adantr 484 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝐴 ∈ On)
198adantr 484 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝐵 ∈ On)
20 simprl 770 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑦𝐵)
21 onelon 6194 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
2219, 20, 21syl2anc 587 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑦 ∈ On)
23 oecl 8149 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
2418, 22, 23syl2anc 587 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → (𝐴o 𝑦) ∈ On)
25 eloni 6179 . . . . . . . . . . 11 ((𝐴o 𝑦) ∈ On → Ord (𝐴o 𝑦))
2624, 25syl 17 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → Ord (𝐴o 𝑦))
27 simprr 772 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑥 ∈ (𝐴o 𝑦))
28 ordsucss 7518 . . . . . . . . . 10 (Ord (𝐴o 𝑦) → (𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ⊆ (𝐴o 𝑦)))
2926, 27, 28sylc 65 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → suc 𝑥 ⊆ (𝐴o 𝑦))
30 simpll 766 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝐴 ∈ (On ∖ 2o))
31 oeordi 8200 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦𝐵 → (𝐴o 𝑦) ∈ (𝐴o 𝐵)))
3219, 30, 31syl2anc 587 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → (𝑦𝐵 → (𝐴o 𝑦) ∈ (𝐴o 𝐵)))
3320, 32mpd 15 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → (𝐴o 𝑦) ∈ (𝐴o 𝐵))
34 onelon 6194 . . . . . . . . . . . 12 (((𝐴o 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴o 𝑦)) → 𝑥 ∈ On)
3524, 27, 34syl2anc 587 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑥 ∈ On)
36 suceloni 7513 . . . . . . . . . . 11 (𝑥 ∈ On → suc 𝑥 ∈ On)
3735, 36syl 17 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → suc 𝑥 ∈ On)
384adantr 484 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → (𝐴o 𝐵) ∈ On)
39 ontr2 6216 . . . . . . . . . 10 ((suc 𝑥 ∈ On ∧ (𝐴o 𝐵) ∈ On) → ((suc 𝑥 ⊆ (𝐴o 𝑦) ∧ (𝐴o 𝑦) ∈ (𝐴o 𝐵)) → suc 𝑥 ∈ (𝐴o 𝐵)))
4037, 38, 39syl2anc 587 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → ((suc 𝑥 ⊆ (𝐴o 𝑦) ∧ (𝐴o 𝑦) ∈ (𝐴o 𝐵)) → suc 𝑥 ∈ (𝐴o 𝐵)))
4129, 33, 40mp2and 698 . . . . . . . 8 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → suc 𝑥 ∈ (𝐴o 𝐵))
4241expr 460 . . . . . . 7 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4317, 42sylan2 595 . . . . . 6 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1o)) → (𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4443rexlimdva 3270 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4516, 44syl5bi 245 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4615, 45sylbid 243 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴o 𝐵) → suc 𝑥 ∈ (𝐴o 𝐵)))
4746ralrimiv 3173 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴o 𝐵)suc 𝑥 ∈ (𝐴o 𝐵))
48 dflim4 7548 . 2 (Lim (𝐴o 𝐵) ↔ (Ord (𝐴o 𝐵) ∧ ∅ ∈ (𝐴o 𝐵) ∧ ∀𝑥 ∈ (𝐴o 𝐵)suc 𝑥 ∈ (𝐴o 𝐵)))
496, 12, 47, 48syl3anbrc 1340 1 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴o 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131   ∖ cdif 3905   ⊆ wss 3908  ∅c0 4265  ∪ ciun 4894  Ord word 6168  Oncon0 6169  Lim wlim 6170  suc csuc 6171  (class class class)co 7140  1oc1o 8082  2oc2o 8083   ↑o coe 8088 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-oexp 8095 This theorem is referenced by:  oaabs2  8259  omabs  8261
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