Theorem oelimcl 8228

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) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑o 𝐵))

Proof of Theorem oelimcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 4105	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2		limelon 6256	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3		oecl 8164	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
4	1, 2, 3	syl2an 597	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) ∈ On)
5		eloni 6203	. . 3 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵))
6	4, 5	syl 17	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Ord (𝐴 ↑o 𝐵))
7	1	adantr 483	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
8	2	adantl 484	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9		dif20el 8132	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
10	9	adantr 483	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11		oen0 8214	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))
12	7, 8, 10, 11	syl21anc 835	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴 ↑o 𝐵))
13		oelim2 8223	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
14	1, 13	sylan 582	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
15	14	eleq2d 2900	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑o 𝐵) ↔ 𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦)))
16		eliun 4925	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴 ↑o 𝑦))
17		eldifi 4105	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ 1o) → 𝑦 ∈ 𝐵)
18	7	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐴 ∈ On)
19	8	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐵 ∈ On)
20		simprl 769	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ 𝐵)
21		onelon 6218	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
22	19, 20, 21	syl2anc 586	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ On)
23		oecl 8164	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
24	18, 22, 23	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝑦) ∈ On)
25		eloni 6203	. . . . . . . . . . 11 ⊢ ((𝐴 ↑o 𝑦) ∈ On → Ord (𝐴 ↑o 𝑦))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → Ord (𝐴 ↑o 𝑦))
27		simprr 771	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ (𝐴 ↑o 𝑦))
28		ordsucss 7535	. . . . . . . . . 10 ⊢ (Ord (𝐴 ↑o 𝑦) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ⊆ (𝐴 ↑o 𝑦)))
29	26, 27, 28	sylc 65	. . . . . . . . 9 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → suc 𝑥 ⊆ (𝐴 ↑o 𝑦))
30		simpll 765	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐴 ∈ (On ∖ 2o))
31		oeordi 8215	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦 ∈ 𝐵 → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o 𝐵)))
32	19, 30, 31	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝑦 ∈ 𝐵 → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o 𝐵)))
33	20, 32	mpd 15	. . . . . . . . 9 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o 𝐵))
34		onelon 6218	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴 ↑o 𝑦)) → 𝑥 ∈ On)
35	24, 27, 34	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ On)
36		suceloni 7530	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
37	35, 36	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → suc 𝑥 ∈ On)
38	4	adantr 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝐵) ∈ On)
39		ontr2 6240	. . . . . . . . . 10 ⊢ ((suc 𝑥 ∈ On ∧ (𝐴 ↑o 𝐵) ∈ On) → ((suc 𝑥 ⊆ (𝐴 ↑o 𝑦) ∧ (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o 𝐵)) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
40	37, 38, 39	syl2anc 586	. . . . . . . . 9 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → ((suc 𝑥 ⊆ (𝐴 ↑o 𝑦) ∧ (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o 𝐵)) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
41	29, 33, 40	mp2and 697	. . . . . . . 8 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → suc 𝑥 ∈ (𝐴 ↑o 𝐵))
42	41	expr 459	. . . . . . 7 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
43	17, 42	sylan2 594	. . . . . 6 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1o)) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
44	43	rexlimdva 3286	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
45	16, 44	syl5bi 244	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
46	15, 45	sylbid 242	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑o 𝐵) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
47	46	ralrimiv 3183	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴 ↑o 𝐵)suc 𝑥 ∈ (𝐴 ↑o 𝐵))
48		dflim4 7565	. 2 ⊢ (Lim (𝐴 ↑o 𝐵) ↔ (Ord (𝐴 ↑o 𝐵) ∧ ∅ ∈ (𝐴 ↑o 𝐵) ∧ ∀𝑥 ∈ (𝐴 ↑o 𝐵)suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
49	6, 12, 47, 48	syl3anbrc 1339	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ∖ cdif 3935   ⊆ wss 3938  ∅c0 4293  ∪ ciun 4921  Ord word 6192  Oncon0 6193  Lim wlim 6194  suc csuc 6195  (class class class)co 7158  1_oc1o 8097  2_oc2o 8098   ↑_o coe 8103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-omul 8109  df-oexp 8110

This theorem is referenced by:  oaabs2  8274  omabs  8276