Theorem oecl 8539

Description: Closure law for ordinal exponentiation. Remark 2.8 of [Schloeder] p. 5. (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 7419	. . . . . . . 8 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8522	. . . . . . . . 9 ⊢ (∅ ↑o ∅) = 1o
3		1on 8480	. . . . . . . . 9 ⊢ 1o ∈ On
4	2, 3	eqeltri 2829	. . . . . . . 8 ⊢ (∅ ↑o ∅) ∈ On
5	1, 4	eqeltrdi 2841	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
6	5	adantl 482	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
7		oe0m1 8523	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
8	7	biimpa 477	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
9		0elon 6418	. . . . . . . 8 ⊢ ∅ ∈ On
10	8, 9	eqeltrdi 2841	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
11	10	adantll 712	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
12	6, 11	oe0lem 8515	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
13	12	anidms 567	. . . 4 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
14		oveq1 7418	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
15	14	eleq1d 2818	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ∈ On ↔ (∅ ↑o 𝐵) ∈ On))
16	13, 15	imbitrrid 245	. . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ On → (𝐴 ↑o 𝐵) ∈ On))
17	16	impcom 408	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) ∈ On)
18		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
19	18	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o ∅) ∈ On))
20		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
21	20	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o 𝑦) ∈ On))
22		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
23	22	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o suc 𝑦) ∈ On))
24		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
25	24	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o 𝐵) ∈ On))
26		oe0 8524	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
27	26, 3	eqeltrdi 2841	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) ∈ On)
28	27	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) ∈ On)
29		omcl 8538	. . . . . . . . . . 11 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On)
30	29	expcom 414	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
31	30	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
32		oesuc 8529	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
33	32	eleq1d 2818	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o suc 𝑦) ∈ On ↔ ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
34	31, 33	sylibrd 258	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On))
35	34	expcom 414	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
36	35	adantrd 492	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
37		vex 3478	. . . . . . . . 9 ⊢ 𝑥 ∈ V
38		iunon 8341	. . . . . . . . 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 8536	. . . . . . . . . . . 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 467	. . . . . . . . . 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 2818	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴 ↑o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On))
45	39, 44	imbitrrid 245	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o 𝑥) ∈ On))
46	45	ex 413	. . . . . 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 7856	. . . . 5 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On))
48	47	expd 416	. . . 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 418	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On)
51	17, 50	oe0lem 8515	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474  ∅c0 4322  ∪ ciun 4997  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7411  1_oc1o 8461   ·_o comu 8466   ↑_o coe 8467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-oexp 8474

This theorem is referenced by:  oen0  8588  oeordi  8589  oeord  8590  oecan  8591  oeword  8592  oewordri  8594  oeworde  8595  oeordsuc  8596  oeoalem  8598  oeoa  8599  oeoelem  8600  oeoe  8601  oelimcl  8602  oeeulem  8603  oeeui  8604  oaabs2  8650  omabs  8652  cantnfle  9668  cantnflt  9669  cantnfp1  9678  cantnflem1d  9685  cantnflem1  9686  cantnflem2  9687  cantnflem3  9688  cantnflem4  9689  cantnf  9690  oemapwe  9691  cantnffval2  9692  cnfcomlem  9696  cnfcom  9697  cnfcom3lem  9700  cnfcom3  9701  infxpenc  10015  onexoegt  42081  oaomoecl  42116  oenassex  42156  cantnftermord  42158  cantnfresb  42162  oacl2g  42168  omabs2  42170  omcl2  42171  ofoaf  42193  ofoafo  42194