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Theorem oelimcl 8614
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 4122 . . . 4 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2 limelon 6427 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3 oecl 8551 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
41, 2, 3syl2an 595 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) ∈ On)
5 eloni 6373 . . 3 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
64, 5syl 17 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴o 𝐵))
71adantr 480 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
82adantl 481 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9 dif20el 8519 . . . 4 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
109adantr 480 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11 oen0 8600 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵))
127, 8, 10, 11syl21anc 837 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴o 𝐵))
13 oelim2 8609 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) = 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦))
141, 13sylan 579 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴o 𝐵) = 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦))
1514eleq2d 2815 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴o 𝐵) ↔ 𝑥 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦)))
16 eliun 4995 . . . . 5 (𝑥 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴o 𝑦))
17 eldifi 4122 . . . . . . 7 (𝑦 ∈ (𝐵 ∖ 1o) → 𝑦𝐵)
187adantr 480 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝐴 ∈ On)
198adantr 480 . . . . . . . . . . . . 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 6388 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
2219, 20, 21syl2anc 583 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑦 ∈ On)
23 oecl 8551 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
2418, 22, 23syl2anc 583 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → (𝐴o 𝑦) ∈ On)
25 eloni 6373 . . . . . . . . . . 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 7815 . . . . . . . . . 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 8601 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦𝐵 → (𝐴o 𝑦) ∈ (𝐴o 𝐵)))
3219, 30, 31syl2anc 583 . . . . . . . . . 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 6388 . . . . . . . . . . . 12 (((𝐴o 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴o 𝑦)) → 𝑥 ∈ On)
3524, 27, 34syl2anc 583 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → 𝑥 ∈ On)
36 onsuc 7808 . . . . . . . . . . 11 (𝑥 ∈ On → suc 𝑥 ∈ On)
3735, 36syl 17 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → suc 𝑥 ∈ On)
384adantr 480 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴o 𝑦))) → (𝐴o 𝐵) ∈ On)
39 ontr2 6410 . . . . . . . . . 10 ((suc 𝑥 ∈ On ∧ (𝐴o 𝐵) ∈ On) → ((suc 𝑥 ⊆ (𝐴o 𝑦) ∧ (𝐴o 𝑦) ∈ (𝐴o 𝐵)) → suc 𝑥 ∈ (𝐴o 𝐵)))
4037, 38, 39syl2anc 583 . . . . . . . . 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 456 . . . . . . 7 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4317, 42sylan2 592 . . . . . 6 (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1o)) → (𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4443rexlimdva 3151 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4516, 44biimtrid 241 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 𝑦 ∈ (𝐵 ∖ 1o)(𝐴o 𝑦) → suc 𝑥 ∈ (𝐴o 𝐵)))
4615, 45sylbid 239 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴o 𝐵) → suc 𝑥 ∈ (𝐴o 𝐵)))
4746ralrimiv 3141 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴o 𝐵)suc 𝑥 ∈ (𝐴o 𝐵))
48 dflim4 7846 . 2 (Lim (𝐴o 𝐵) ↔ (Ord (𝐴o 𝐵) ∧ ∅ ∈ (𝐴o 𝐵) ∧ ∀𝑥 ∈ (𝐴o 𝐵)suc 𝑥 ∈ (𝐴o 𝐵)))
496, 12, 47, 48syl3anbrc 1341 1 ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3057  wrex 3066  cdif 3942  wss 3945  c0 4318   ciun 4991  Ord word 6362  Oncon0 6363  Lim wlim 6364  suc csuc 6365  (class class class)co 7414  1oc1o 8473  2oc2o 8474  o coe 8479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-oexp 8486
This theorem is referenced by:  oaabs2  8663  omabs  8665  rp-oelim2  42731
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