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Theorem oelimcl 7851
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 3875 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
2 limelon 5949 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3 oecl 7788 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
41, 2, 3syl2an 495 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) ∈ On)
5 eloni 5894 . . 3 ((𝐴𝑜 𝐵) ∈ On → Ord (𝐴𝑜 𝐵))
64, 5syl 17 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴𝑜 𝐵))
71adantr 472 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
82adantl 473 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9 dif20el 7756 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
109adantr 472 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11 oen0 7837 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝐵))
127, 8, 10, 11syl21anc 1476 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴𝑜 𝐵))
13 oelim2 7846 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) = 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦))
141, 13sylan 489 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) = 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦))
1514eleq2d 2825 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴𝑜 𝐵) ↔ 𝑥 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦)))
16 eliun 4676 . . . . 5 (𝑥 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1𝑜)𝑥 ∈ (𝐴𝑜 𝑦))
17 eldifi 3875 . . . . . . 7 (𝑦 ∈ (𝐵 ∖ 1𝑜) → 𝑦𝐵)
187adantr 472 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝐴 ∈ On)
198adantr 472 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝐵 ∈ On)
20 simprl 811 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑦𝐵)
21 onelon 5909 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
2219, 20, 21syl2anc 696 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑦 ∈ On)
23 oecl 7788 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
2418, 22, 23syl2anc 696 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → (𝐴𝑜 𝑦) ∈ On)
25 eloni 5894 . . . . . . . . . . 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 813 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑥 ∈ (𝐴𝑜 𝑦))
28 ordsucss 7184 . . . . . . . . . 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 807 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝐴 ∈ (On ∖ 2𝑜))
31 oeordi 7838 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑦𝐵 → (𝐴𝑜 𝑦) ∈ (𝐴𝑜 𝐵)))
3219, 30, 31syl2anc 696 . . . . . . . . . 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 5909 . . . . . . . . . . . 12 (((𝐴𝑜 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴𝑜 𝑦)) → 𝑥 ∈ On)
3524, 27, 34syl2anc 696 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑥 ∈ On)
36 suceloni 7179 . . . . . . . . . . 11 (𝑥 ∈ On → suc 𝑥 ∈ On)
3735, 36syl 17 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → suc 𝑥 ∈ On)
384adantr 472 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → (𝐴𝑜 𝐵) ∈ On)
39 ontr2 5933 . . . . . . . . . 10 ((suc 𝑥 ∈ On ∧ (𝐴𝑜 𝐵) ∈ On) → ((suc 𝑥 ⊆ (𝐴𝑜 𝑦) ∧ (𝐴𝑜 𝑦) ∈ (𝐴𝑜 𝐵)) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4037, 38, 39syl2anc 696 . . . . . . . . 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 717 . . . . . . . 8 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → suc 𝑥 ∈ (𝐴𝑜 𝐵))
4241expr 644 . . . . . . 7 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4317, 42sylan2 492 . . . . . 6 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1𝑜)) → (𝑥 ∈ (𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4443rexlimdva 3169 . . . . 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 3103 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴𝑜 𝐵)suc 𝑥 ∈ (𝐴𝑜 𝐵))
48 dflim4 7214 . 2 (Lim (𝐴𝑜 𝐵) ↔ (Ord (𝐴𝑜 𝐵) ∧ ∅ ∈ (𝐴𝑜 𝐵) ∧ ∀𝑥 ∈ (𝐴𝑜 𝐵)suc 𝑥 ∈ (𝐴𝑜 𝐵)))
496, 12, 47, 48syl3anbrc 1429 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cdif 3712  wss 3715  c0 4058   ciun 4672  Ord word 5883  Oncon0 5884  Lim wlim 5885  suc csuc 5886  (class class class)co 6814  1𝑜c1o 7723  2𝑜c2o 7724  𝑜 coe 7729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-omul 7735  df-oexp 7736
This theorem is referenced by:  oaabs2  7896  omabs  7898
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