Theorem oecl 8522

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 8505	. . . . . . . . 9 ⊢ (∅ ↑o ∅) = 1o
3		1on 8466	. . . . . . . . 9 ⊢ 1o ∈ On
4	2, 3	eqeltri 2865	. . . . . . . 8 ⊢ (∅ ↑o ∅) ∈ On
5	1, 4	eqeltrdi 2877	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
6	5	adantl 486	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
7		oe0m1 8506	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
8	7	biimpa 481	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
9		0elon 6417	. . . . . . . 8 ⊢ ∅ ∈ On
10	8, 9	eqeltrdi 2877	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
11	10	adantll 726	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
12	6, 11	oe0lem 8498	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
13	12	anidms 576	. . . 4 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
14		oveq1 7418	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
15	14	eleq1d 2854	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ∈ On ↔ (∅ ↑o 𝐵) ∈ On))
16	13, 15	imbitrrid 249	. . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ On → (𝐴 ↑o 𝐵) ∈ On))
17	16	impcom 412	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) ∈ On)
18		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
19	18	eleq1d 2854	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o ∅) ∈ On))
20		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
21	20	eleq1d 2854	. . . . . 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 2854	. . . . . 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 2854	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o 𝐵) ∈ On))
26		oe0 8507	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
27	26, 3	eqeltrdi 2877	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) ∈ On)
28	27	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) ∈ On)
29		omcl 8521	. . . . . . . . . . 11 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On)
30	29	expcom 418	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
31	30	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
32		oesuc 8512	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
33	32	eleq1d 2854	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o suc 𝑦) ∈ On ↔ ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
34	31, 33	sylibrd 262	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On))
35	34	expcom 418	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
36	35	adantrd 496	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
37		vex 3467	. . . . . . . . 9 ⊢ 𝑥 ∈ V
38		iunon 8326	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On)
39	37, 38	mpan 702	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On)
40		oelim 8519	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
41	37, 40	mpanlr1 718	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
42	41	anasss 471	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43	42	an12s 661	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
44	43	eleq1d 2854	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴 ↑o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On))
45	39, 44	imbitrrid 249	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o 𝑥) ∈ On))
46	45	ex 417	. . . . . 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 7861	. . . . 5 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On))
48	47	expd 420	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (𝐴 ↑o 𝐵) ∈ On)))
49	48	com12 33	. . 3 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → (𝐴 ↑o 𝐵) ∈ On)))
50	49	imp31 422	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On)
51	17, 50	oe0lem 8498	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463  ∅c0 4294  ∪ ciun 4960  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7411  1_oc1o 8446   ·_o comu 8451   ↑_o coe 8452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-omul 8458  df-oexp 8459

This theorem is referenced by:  oen0  8572  oeordi  8573  oeord  8574  oecan  8575  oeword  8576  oewordri  8578  oeworde  8579  oeordsuc  8580  oeoalem  8582  oeoa  8583  oeoelem  8584  oeoe  8585  oelimcl  8586  oeeulem  8587  oeeui  8588  oaabs2  8635  omabs  8637  cantnfle  9640  cantnflt  9641  cantnfp1  9650  cantnflem1d  9657  cantnflem1  9658  cantnflem2  9659  cantnflem3  9660  cantnflem4  9661  cantnf  9662  oemapwe  9663  cantnffval2  9664  cnfcomlem  9668  cnfcom  9669  cnfcom3lem  9672  cnfcom3  9673  infxpenc  10002  onexoegt  43863  oaomoecl  43897  oenassex  43937  cantnftermord  43939  cantnfresb  43943  oacl2g  43949  omabs2  43951  omcl2  43952  ofoaf  43974  ofoafo  43975