Theorem oelimcl 7680

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 3732	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
2		limelon 5788	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3		oecl 7617	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
4	1, 2, 3	syl2an 494	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑𝑜 𝐵) ∈ On)
5		eloni 5733	. . 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 7585	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
10	9	adantr 481	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11		oen0 7666	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝐵))
12	7, 8, 10, 11	syl21anc 1325	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴 ↑𝑜 𝐵))
13		oelim2 7675	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦))
14	1, 13	sylan 488	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦))
15	14	eleq2d 2687	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑𝑜 𝐵) ↔ 𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦)))
16		eliun 4524	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1𝑜)𝑥 ∈ (𝐴 ↑𝑜 𝑦))
17		eldifi 3732	. . . . . . 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 794	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝑦 ∈ 𝐵)
21		onelon 5748	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
22	19, 20, 21	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝑦 ∈ On)
23		oecl 7617	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
24	18, 22, 23	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → (𝐴 ↑𝑜 𝑦) ∈ On)
25		eloni 5733	. . . . . . . . . . 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 796	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝑥 ∈ (𝐴 ↑𝑜 𝑦))
28		ordsucss 7018	. . . . . . . . . 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 790	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝐴 ∈ (On ∖ 2𝑜))
31		oeordi 7667	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑦 ∈ 𝐵 → (𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 𝐵)))
32	19, 30, 31	syl2anc 693	. . . . . . . . . 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 5748	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑𝑜 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦)) → 𝑥 ∈ On)
35	24, 27, 34	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝑥 ∈ On)
36		suceloni 7013	. . . . . . . . . . 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 5772	. . . . . . . . . 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 693	. . . . . . . . 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 715	. . . . . . . 8 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵))
42	41	expr 643	. . . . . . 7 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ↑𝑜 𝑦) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
43	17, 42	sylan2 491	. . . . . 6 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1𝑜)) → (𝑥 ∈ (𝐴 ↑𝑜 𝑦) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
44	43	rexlimdva 3031	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1𝑜)𝑥 ∈ (𝐴 ↑𝑜 𝑦) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
45	16, 44	syl5bi 232	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
46	15, 45	sylbid 230	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑𝑜 𝐵) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
47	46	ralrimiv 2965	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴 ↑𝑜 𝐵)suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵))
48		dflim4 7048	. 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 1246	1 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑𝑜 𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oaabs2  7725  omabs  7727