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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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