Theorem oelimcl 8526

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 4061	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2		limelon 6375	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3		oecl 8462	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
4	1, 2, 3	syl2an 602	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) ∈ On)
5		eloni 6320	. . 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 481	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
8	2	adantl 482	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9		dif20el 8430	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
10	9	adantr 481	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11		oen0 8512	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))
12	7, 8, 10, 11	syl21anc 843	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴 ↑o 𝐵))
13		oelim2 8521	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
14	1, 13	sylan 586	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
15	14	eleq2d 2825	. . . 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 4061	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ 1o) → 𝑦 ∈ 𝐵)
18	7	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐴 ∈ On)
19	8	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐵 ∈ On)
20		simprl 776	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ 𝐵)
21		onelon 6335	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
22	19, 20, 21	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ On)
23		oecl 8462	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
24	18, 22, 23	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝑦) ∈ On)
25		eloni 6320	. . . . . . . . . . 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 778	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ (𝐴 ↑o 𝑦))
28		ordsucss 7758	. . . . . . . . . 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 772	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐴 ∈ (On ∖ 2o))
31		oeordi 8513	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦 ∈ 𝐵 → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o 𝐵)))
32	19, 30, 31	syl2anc 590	. . . . . . . . . 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 6335	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴 ↑o 𝑦)) → 𝑥 ∈ On)
35	24, 27, 34	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ On)
36		onsuc 7753	. . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝐵) ∈ On)
39		ontr2 6358	. . . . . . . . . 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 590	. . . . . . . . 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 705	. . . . . . . 8 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → suc 𝑥 ∈ (𝐴 ↑o 𝐵))
42	41	expr 457	. . . . . . 7 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
43	17, 42	sylan2 599	. . . . . 6 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1o)) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
44	43	rexlimdva 3140	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
45	16, 44	biimtrid 243	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
46	15, 45	sylbid 241	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑o 𝐵) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
47	46	ralrimiv 3130	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴 ↑o 𝐵)suc 𝑥 ∈ (𝐴 ↑o 𝐵))
48		dflim4 7788	. 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 1350	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4261  ∪ ciun 4921  Ord word 6309  Oncon0 6310  Lim wlim 6311  suc csuc 6312  (class class class)co 7356  1_oc1o 8388  2_oc2o 8389   ↑_o coe 8394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-omul 8400  df-oexp 8401

This theorem is referenced by:  oaabs2  8575  omabs  8577  rp-oelim2  43753