Theorem oelimcl 8612

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 4106	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2		limelon 6417	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3		oecl 8549	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
4	1, 2, 3	syl2an 596	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) ∈ On)
5		eloni 6362	. . 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 480	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
8	2	adantl 481	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9		dif20el 8517	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
10	9	adantr 480	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11		oen0 8598	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))
12	7, 8, 10, 11	syl21anc 837	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴 ↑o 𝐵))
13		oelim2 8607	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
14	1, 13	sylan 580	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
15	14	eleq2d 2820	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑o 𝐵) ↔ 𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦)))
16		eliun 4971	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴 ↑o 𝑦))
17		eldifi 4106	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ 1o) → 𝑦 ∈ 𝐵)
18	7	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐴 ∈ On)
19	8	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐵 ∈ On)
20		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ 𝐵)
21		onelon 6377	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
22	19, 20, 21	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ On)
23		oecl 8549	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
24	18, 22, 23	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝑦) ∈ On)
25		eloni 6362	. . . . . . . . . . 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 772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ (𝐴 ↑o 𝑦))
28		ordsucss 7812	. . . . . . . . . 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 766	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐴 ∈ (On ∖ 2o))
31		oeordi 8599	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦 ∈ 𝐵 → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o 𝐵)))
32	19, 30, 31	syl2anc 584	. . . . . . . . . 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 6377	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴 ↑o 𝑦)) → 𝑥 ∈ On)
35	24, 27, 34	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ On)
36		onsuc 7805	. . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝐵) ∈ On)
39		ontr2 6400	. . . . . . . . . 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 584	. . . . . . . . 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 699	. . . . . . . 8 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → suc 𝑥 ∈ (𝐴 ↑o 𝐵))
42	41	expr 456	. . . . . . 7 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
43	17, 42	sylan2 593	. . . . . 6 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1o)) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
44	43	rexlimdva 3141	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
45	16, 44	biimtrid 242	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
46	15, 45	sylbid 240	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑o 𝐵) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
47	46	ralrimiv 3131	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴 ↑o 𝐵)suc 𝑥 ∈ (𝐴 ↑o 𝐵))
48		dflim4 7843	. 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 1344	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  ∪ ciun 4967  Ord word 6351  Oncon0 6352  Lim wlim 6353  suc csuc 6354  (class class class)co 7405  1_oc1o 8473  2_oc2o 8474   ↑_o coe 8479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  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  8661  omabs  8663  rp-oelim2  43332