Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oelimcl Structured version   Visualization version   GIF version

Theorem oelimcl 7567
 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 3694 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
2 limelon 5705 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3 oecl 7504 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
41, 2, 3syl2an 493 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) ∈ On)
5 eloni 5650 . . 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 7472 . . . 4 (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
109adantr 480 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11 oen0 7553 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝐵))
127, 8, 10, 11syl21anc 1317 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴𝑜 𝐵))
13 oelim2 7562 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) = 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦))
141, 13sylan 487 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) = 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦))
1514eleq2d 2673 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴𝑜 𝐵) ↔ 𝑥 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦)))
16 eliun 4460 . . . . 5 (𝑥 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1𝑜)𝑥 ∈ (𝐴𝑜 𝑦))
17 eldifi 3694 . . . . . . 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 790 . . . . . . . . . . . . 13 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑦𝐵)
21 onelon 5665 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
2219, 20, 21syl2anc 691 . . . . . . . . . . . 12 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑦 ∈ On)
23 oecl 7504 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
2418, 22, 23syl2anc 691 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → (𝐴𝑜 𝑦) ∈ On)
25 eloni 5650 . . . . . . . . . . 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 792 . . . . . . . . . 10 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑥 ∈ (𝐴𝑜 𝑦))
28 ordsucss 6910 . . . . . . . . . 10 (Ord (𝐴𝑜 𝑦) → (𝑥 ∈ (𝐴𝑜 𝑦) → suc 𝑥 ⊆ (𝐴𝑜 𝑦)))
2926, 27, 28sylc 63 . . . . . . . . 9 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → suc 𝑥 ⊆ (𝐴𝑜 𝑦))
30 simpll 786 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝐴 ∈ (On ∖ 2𝑜))
31 oeordi 7554 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑦𝐵 → (𝐴𝑜 𝑦) ∈ (𝐴𝑜 𝐵)))
3219, 30, 31syl2anc 691 . . . . . . . . . 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 5665 . . . . . . . . . . . 12 (((𝐴𝑜 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴𝑜 𝑦)) → 𝑥 ∈ On)
3524, 27, 34syl2anc 691 . . . . . . . . . . 11 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → 𝑥 ∈ On)
36 suceloni 6905 . . . . . . . . . . 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 5689 . . . . . . . . . 10 ((suc 𝑥 ∈ On ∧ (𝐴𝑜 𝐵) ∈ On) → ((suc 𝑥 ⊆ (𝐴𝑜 𝑦) ∧ (𝐴𝑜 𝑦) ∈ (𝐴𝑜 𝐵)) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4037, 38, 39syl2anc 691 . . . . . . . . 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 711 . . . . . . . 8 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝑦𝐵𝑥 ∈ (𝐴𝑜 𝑦))) → suc 𝑥 ∈ (𝐴𝑜 𝐵))
4241expr 641 . . . . . . 7 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4317, 42sylan2 490 . . . . . 6 (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1𝑜)) → (𝑥 ∈ (𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4443rexlimdva 3013 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1𝑜)𝑥 ∈ (𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4516, 44syl5bi 231 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑦) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4615, 45sylbid 229 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴𝑜 𝐵) → suc 𝑥 ∈ (𝐴𝑜 𝐵)))
4746ralrimiv 2948 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴𝑜 𝐵)suc 𝑥 ∈ (𝐴𝑜 𝐵))
48 dflim4 6940 . 2 (Lim (𝐴𝑜 𝐵) ↔ (Ord (𝐴𝑜 𝐵) ∧ ∅ ∈ (𝐴𝑜 𝐵) ∧ ∀𝑥 ∈ (𝐴𝑜 𝐵)suc 𝑥 ∈ (𝐴𝑜 𝐵)))
496, 12, 47, 48syl3anbrc 1239 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴𝑜 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1𝑜c1o 7440  2𝑜c2o 7441   ↑𝑜 coe 7446 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453 This theorem is referenced by:  oaabs2  7612  omabs  7614
 Copyright terms: Public domain W3C validator