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Theorem oelimcl 7625
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 3710 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
2 limelon 5747 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3 oecl 7562 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
41, 2, 3syl2an 494 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) ∈ On)
5 eloni 5692 . . 3 ((𝐴𝑜 𝐵) ∈ On → Ord (𝐴𝑜 𝐵))
64, 5syl 17 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴𝑜 𝐵))
71adantr 481 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
82adantl 482 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9 dif20el 7530 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
109adantr 481 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11 oen0 7611 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝐵))
127, 8, 10, 11syl21anc 1322 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴𝑜 𝐵))
13 oelim2 7620 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) = 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦))
141, 13sylan 488 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) = 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦))
1514eleq2d 2684 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴𝑜 𝐵) ↔ 𝑥 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦)))
16 eliun 4490 . . . . 5 (𝑥 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1𝑜)𝑥 ∈ (𝐴𝑜 𝑦))
17 eldifi 3710 . . . . . . 7 (𝑦 ∈ (𝐵 ∖ 1𝑜) → 𝑦𝐵)
187adantr 481 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝐴 ∈ On)
198adantr 481 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝐵 ∈ On)
20 simprl 793 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑦𝐵)
21 onelon 5707 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
2219, 20, 21syl2anc 692 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑦 ∈ On)
23 oecl 7562 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
2418, 22, 23syl2anc 692 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → (𝐴𝑜 𝑦) ∈ On)
25 eloni 5692 . . . . . . . . . . 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 795 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑥 ∈ (𝐴𝑜 𝑦))
28 ordsucss 6965 . . . . . . . . . 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 789 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝐴 ∈ (On ∖ 2𝑜))
31 oeordi 7612 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑦𝐵 → (𝐴𝑜 𝑦) ∈ (𝐴𝑜 𝐵)))
3219, 30, 31syl2anc 692 . . . . . . . . . 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 5707 . . . . . . . . . . . 12 (((𝐴𝑜 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴𝑜 𝑦)) → 𝑥 ∈ On)
3524, 27, 34syl2anc 692 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑥 ∈ On)
36 suceloni 6960 . . . . . . . . . . 11 (𝑥 ∈ On → suc 𝑥 ∈ On)
3735, 36syl 17 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → suc 𝑥 ∈ On)
384adantr 481 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → (𝐴𝑜 𝐵) ∈ On)
39 ontr2 5731 . . . . . . . . . 10 ((suc 𝑥 ∈ On ∧ (𝐴𝑜 𝐵) ∈ On) → ((suc 𝑥 ⊆ (𝐴𝑜 𝑦) ∧ (𝐴𝑜 𝑦) ∈ (𝐴𝑜 𝐵)) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4037, 38, 39syl2anc 692 . . . . . . . . 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 714 . . . . . . . 8 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → suc 𝑥 ∈ (𝐴𝑜 𝐵))
4241expr 642 . . . . . . 7 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4317, 42sylan2 491 . . . . . 6 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1𝑜)) → (𝑥 ∈ (𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4443rexlimdva 3024 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1𝑜)𝑥 ∈ (𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4516, 44syl5bi 232 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4615, 45sylbid 230 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴𝑜 𝐵) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4746ralrimiv 2959 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴𝑜 𝐵)suc 𝑥 ∈ (𝐴𝑜 𝐵))
48 dflim4 6995 . 2 (Lim (𝐴𝑜 𝐵) ↔ (Ord (𝐴𝑜 𝐵) ∧ ∅ ∈ (𝐴𝑜 𝐵) ∧ ∀𝑥 ∈ (𝐴𝑜 𝐵)suc 𝑥 ∈ (𝐴𝑜 𝐵)))
496, 12, 47, 48syl3anbrc 1244 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cdif 3552  wss 3555  c0 3891   ciun 4485  Ord word 5681  Oncon0 5682  Lim wlim 5683  suc csuc 5684  (class class class)co 6604  1𝑜c1o 7498  2𝑜c2o 7499  𝑜 coe 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511
This theorem is referenced by:  oaabs2  7670  omabs  7672
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