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Theorem oelimcl 7964
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 3955 . . . 4 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2 limelon 6039 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3 oecl 7901 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
41, 2, 3syl2an 589 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) ∈ On)
5 eloni 5986 . . 3 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
64, 5syl 17 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴o 𝐵))
71adantr 474 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
82adantl 475 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9 dif20el 7869 . . . 4 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
109adantr 474 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11 oen0 7950 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵))
127, 8, 10, 11syl21anc 828 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴o 𝐵))
13 oelim2 7959 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) = 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦))
141, 13sylan 575 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) = 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦))
1514eleq2d 2845 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴o 𝐵) ↔ 𝑥 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦)))
16 eliun 4757 . . . . 5 (𝑥 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴o 𝑦))
17 eldifi 3955 . . . . . . 7 (𝑦 ∈ (𝐵 ∖ 1o) → 𝑦𝐵)
187adantr 474 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝐴 ∈ On)
198adantr 474 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝐵 ∈ On)
20 simprl 761 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑦𝐵)
21 onelon 6001 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
2219, 20, 21syl2anc 579 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑦 ∈ On)
23 oecl 7901 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
2418, 22, 23syl2anc 579 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → (𝐴o 𝑦) ∈ On)
25 eloni 5986 . . . . . . . . . . 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 763 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑥 ∈ (𝐴o 𝑦))
28 ordsucss 7296 . . . . . . . . . 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 757 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝐴 ∈ (On ∖ 2o))
31 oeordi 7951 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦𝐵 → (𝐴o 𝑦) ∈ (𝐴o 𝐵)))
3219, 30, 31syl2anc 579 . . . . . . . . . 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 6001 . . . . . . . . . . . 12 (((𝐴o 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴o 𝑦)) → 𝑥 ∈ On)
3524, 27, 34syl2anc 579 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑥 ∈ On)
36 suceloni 7291 . . . . . . . . . . 11 (𝑥 ∈ On → suc 𝑥 ∈ On)
3735, 36syl 17 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → suc 𝑥 ∈ On)
384adantr 474 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → (𝐴o 𝐵) ∈ On)
39 ontr2 6023 . . . . . . . . . 10 ((suc 𝑥 ∈ On ∧ (𝐴o 𝐵) ∈ On) → ((suc 𝑥 ⊆ (𝐴o 𝑦) ∧ (𝐴o 𝑦) ∈ (𝐴o 𝐵)) → suc 𝑥 ∈ (𝐴o 𝐵)))
4037, 38, 39syl2anc 579 . . . . . . . . 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 689 . . . . . . . 8 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → suc 𝑥 ∈ (𝐴o 𝐵))
4241expr 450 . . . . . . 7 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4317, 42sylan2 586 . . . . . 6 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1o)) → (𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4443rexlimdva 3213 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4516, 44syl5bi 234 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4615, 45sylbid 232 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴o 𝐵) → suc 𝑥 ∈ (𝐴o 𝐵)))
4746ralrimiv 3147 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴o 𝐵)suc 𝑥 ∈ (𝐴o 𝐵))
48 dflim4 7326 . 2 (Lim (𝐴o 𝐵) ↔ (Ord (𝐴o 𝐵) ∧ ∅ ∈ (𝐴o 𝐵) ∧ ∀𝑥 ∈ (𝐴o 𝐵)suc 𝑥 ∈ (𝐴o 𝐵)))
496, 12, 47, 48syl3anbrc 1400 1 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  cdif 3789  wss 3792  c0 4141   ciun 4753  Ord word 5975  Oncon0 5976  Lim wlim 5977  suc csuc 5978  (class class class)co 6922  1oc1o 7836  2oc2o 7837  o coe 7842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-omul 7848  df-oexp 7849
This theorem is referenced by:  oaabs2  8009  omabs  8011
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