Theorem oecl 8575

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 7439	. . . . . . . 8 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8558	. . . . . . . . 9 ⊢ (∅ ↑o ∅) = 1o
3		1on 8518	. . . . . . . . 9 ⊢ 1o ∈ On
4	2, 3	eqeltri 2837	. . . . . . . 8 ⊢ (∅ ↑o ∅) ∈ On
5	1, 4	eqeltrdi 2849	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
6	5	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
7		oe0m1 8559	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
8	7	biimpa 476	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
9		0elon 6438	. . . . . . . 8 ⊢ ∅ ∈ On
10	8, 9	eqeltrdi 2849	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
11	10	adantll 714	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
12	6, 11	oe0lem 8551	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
13	12	anidms 566	. . . 4 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
14		oveq1 7438	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
15	14	eleq1d 2826	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ∈ On ↔ (∅ ↑o 𝐵) ∈ On))
16	13, 15	imbitrrid 246	. . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ On → (𝐴 ↑o 𝐵) ∈ On))
17	16	impcom 407	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) ∈ On)
18		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
19	18	eleq1d 2826	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o ∅) ∈ On))
20		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
21	20	eleq1d 2826	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o 𝑦) ∈ On))
22		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
23	22	eleq1d 2826	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o suc 𝑦) ∈ On))
24		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
25	24	eleq1d 2826	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o 𝐵) ∈ On))
26		oe0 8560	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
27	26, 3	eqeltrdi 2849	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) ∈ On)
28	27	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) ∈ On)
29		omcl 8574	. . . . . . . . . . 11 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On)
30	29	expcom 413	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
31	30	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
32		oesuc 8565	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
33	32	eleq1d 2826	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o suc 𝑦) ∈ On ↔ ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
34	31, 33	sylibrd 259	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On))
35	34	expcom 413	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
36	35	adantrd 491	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
37		vex 3484	. . . . . . . . 9 ⊢ 𝑥 ∈ V
38		iunon 8379	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On)
39	37, 38	mpan 690	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On)
40		oelim 8572	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
41	37, 40	mpanlr1 706	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
42	41	anasss 466	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43	42	an12s 649	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
44	43	eleq1d 2826	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴 ↑o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On))
45	39, 44	imbitrrid 246	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o 𝑥) ∈ On))
46	45	ex 412	. . . . . 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 7886	. . . . 5 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On))
48	47	expd 415	. . . 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 417	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On)
51	17, 50	oe0lem 8551	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480  ∅c0 4333  ∪ ciun 4991  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431  1_oc1o 8499   ·_o comu 8504   ↑_o coe 8505

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-oexp 8512

This theorem is referenced by:  oen0  8624  oeordi  8625  oeord  8626  oecan  8627  oeword  8628  oewordri  8630  oeworde  8631  oeordsuc  8632  oeoalem  8634  oeoa  8635  oeoelem  8636  oeoe  8637  oelimcl  8638  oeeulem  8639  oeeui  8640  oaabs2  8687  omabs  8689  cantnfle  9711  cantnflt  9712  cantnfp1  9721  cantnflem1d  9728  cantnflem1  9729  cantnflem2  9730  cantnflem3  9731  cantnflem4  9732  cantnf  9733  oemapwe  9734  cantnffval2  9735  cnfcomlem  9739  cnfcom  9740  cnfcom3lem  9743  cnfcom3  9744  infxpenc  10058  onexoegt  43256  oaomoecl  43291  oenassex  43331  cantnftermord  43333  cantnfresb  43337  oacl2g  43343  omabs2  43345  omcl2  43346  ofoaf  43368  ofoafo  43369