Theorem oecl 8161

Description: Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
oecl	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Proof of Theorem oecl

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

Step	Hyp	Ref	Expression
1		oveq2 7163	. . . . . . . 8 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8144	. . . . . . . . 9 ⊢ (∅ ↑o ∅) = 1o
3		1on 8108	. . . . . . . . 9 ⊢ 1o ∈ On
4	2, 3	eqeltri 2909	. . . . . . . 8 ⊢ (∅ ↑o ∅) ∈ On
5	1, 4	eqeltrdi 2921	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
6	5	adantl 484	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
7		oe0m1 8145	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
8	7	biimpa 479	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
9		0elon 6243	. . . . . . . 8 ⊢ ∅ ∈ On
10	8, 9	eqeltrdi 2921	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
11	10	adantll 712	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
12	6, 11	oe0lem 8137	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
13	12	anidms 569	. . . 4 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
14		oveq1 7162	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
15	14	eleq1d 2897	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ∈ On ↔ (∅ ↑o 𝐵) ∈ On))
16	13, 15	syl5ibr 248	. . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ On → (𝐴 ↑o 𝐵) ∈ On))
17	16	impcom 410	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) ∈ On)
18		oveq2 7163	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
19	18	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o ∅) ∈ On))
20		oveq2 7163	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
21	20	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o 𝑦) ∈ On))
22		oveq2 7163	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
23	22	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o suc 𝑦) ∈ On))
24		oveq2 7163	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
25	24	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o 𝐵) ∈ On))
26		oe0 8146	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
27	26, 3	eqeltrdi 2921	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) ∈ On)
28	27	adantr 483	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) ∈ On)
29		omcl 8160	. . . . . . . . . . 11 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On)
30	29	expcom 416	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
31	30	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
32		oesuc 8151	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
33	32	eleq1d 2897	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o suc 𝑦) ∈ On ↔ ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
34	31, 33	sylibrd 261	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On))
35	34	expcom 416	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
36	35	adantrd 494	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
37		vex 3497	. . . . . . . . 9 ⊢ 𝑥 ∈ V
38		iunon 7975	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On)
39	37, 38	mpan 688	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On)
40		oelim 8158	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
41	37, 40	mpanlr1 704	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
42	41	anasss 469	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43	42	an12s 647	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
44	43	eleq1d 2897	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴 ↑o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On))
45	39, 44	syl5ibr 248	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o 𝑥) ∈ On))
46	45	ex 415	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o 𝑥) ∈ On)))
47	19, 21, 23, 25, 28, 36, 46	tfinds3 7578	. . . . 5 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On))
48	47	expd 418	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (𝐴 ↑o 𝐵) ∈ On)))
49	48	com12 32	. . 3 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝐴 ↑o 𝐵) ∈ On)))
50	49	imp31 420	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On)
51	17, 50	oe0lem 8137	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494  ∅c0 4290  ∪ ciun 4918  Oncon0 6190  Lim wlim 6191  suc csuc 6192  (class class class)co 7155  1_oc1o 8094   ·_o comu 8099   ↑_o coe 8100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-omul 8106  df-oexp 8107

This theorem is referenced by:  oen0  8211  oeordi  8212  oeord  8213  oecan  8214  oeword  8215  oewordri  8217  oeworde  8218  oeordsuc  8219  oeoalem  8221  oeoa  8222  oeoelem  8223  oeoe  8224  oelimcl  8225  oeeulem  8226  oeeui  8227  oaabs2  8271  omabs  8273  cantnfle  9133  cantnflt  9134  cantnfp1  9143  cantnflem1d  9150  cantnflem1  9151  cantnflem2  9152  cantnflem3  9153  cantnflem4  9154  cantnf  9155  oemapwe  9156  cantnffval2  9157  cnfcomlem  9161  cnfcom  9162  cnfcom3lem  9165  cnfcom3  9166  infxpenc  9443