Theorem oecl 7504

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 6557	. . . . . . . 8 ⊢ (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2		oe0m0 7487	. . . . . . . . 9 ⊢ (∅ ↑𝑜 ∅) = 1𝑜
3		1on 7454	. . . . . . . . 9 ⊢ 1𝑜 ∈ On
4	2, 3	eqeltri 2684	. . . . . . . 8 ⊢ (∅ ↑𝑜 ∅) ∈ On
5	1, 4	syl6eqel 2696	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅ ↑𝑜 𝐵) ∈ On)
6	5	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑𝑜 𝐵) ∈ On)
7		oe0m1 7488	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
8	7	biimpa 500	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
9		0elon 5695	. . . . . . . 8 ⊢ ∅ ∈ On
10	8, 9	syl6eqel 2696	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
11	10	adantll 746	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
12	6, 11	oe0lem 7480	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
13	12	anidms 675	. . . 4 ⊢ (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
14		oveq1 6556	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
15	14	eleq1d 2672	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) ∈ On ↔ (∅ ↑𝑜 𝐵) ∈ On))
16	13, 15	syl5ibr 235	. . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ On → (𝐴 ↑𝑜 𝐵) ∈ On))
17	16	impcom 445	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 𝐵) ∈ On)
18		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
19	18	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑜 𝑥) ∈ On ↔ (𝐴 ↑𝑜 ∅) ∈ On))
20		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
21	20	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑜 𝑥) ∈ On ↔ (𝐴 ↑𝑜 𝑦) ∈ On))
22		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
23	22	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝑥) ∈ On ↔ (𝐴 ↑𝑜 suc 𝑦) ∈ On))
24		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐵))
25	24	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑𝑜 𝑥) ∈ On ↔ (𝐴 ↑𝑜 𝐵) ∈ On))
26		oe0 7489	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
27	26, 3	syl6eqel 2696	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) ∈ On)
28	27	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) ∈ On)
29		omcl 7503	. . . . . . . . . . 11 ⊢ (((𝐴 ↑𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ On)
30	29	expcom 450	. . . . . . . . . 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 7494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
33	32	eleq1d 2672	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑𝑜 suc 𝑦) ∈ On ↔ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ On))
34	31, 33	sylibrd 248	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑𝑜 𝑦) ∈ On → (𝐴 ↑𝑜 suc 𝑦) ∈ On))
35	34	expcom 450	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ↑𝑜 𝑦) ∈ On → (𝐴 ↑𝑜 suc 𝑦) ∈ On)))
36	35	adantrd 483	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴 ↑𝑜 𝑦) ∈ On → (𝐴 ↑𝑜 suc 𝑦) ∈ On)))
37		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
38		iunon 7323	. . . . . . . . 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 7501	. . . . . . . . . . . 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 677	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
43	42	an12s 839	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
44	43	eleq1d 2672	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴 ↑𝑜 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ∈ On))
45	39, 44	syl5ibr 235	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ∈ On → (𝐴 ↑𝑜 𝑥) ∈ On))
46	45	ex 449	. . . . . 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 6956	. . . . 5 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) ∈ On))
48	47	expd 451	. . . 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 447	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) ∈ On)
51	17, 50	oe0lem 7480	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∅c0 3874  ∪ ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  oen0  7553  oeordi  7554  oeord  7555  oecan  7556  oeword  7557  oewordri  7559  oeworde  7560  oeordsuc  7561  oeoalem  7563  oeoa  7564  oeoelem  7565  oeoe  7566  oelimcl  7567  oeeulem  7568  oeeui  7569  oaabs2  7612  omabs  7614  cantnfle  8451  cantnflt  8452  cantnfp1  8461  cantnflem1d  8468  cantnflem1  8469  cantnflem2  8470  cantnflem3  8471  cantnflem4  8472  cantnf  8473  oemapwe  8474  cantnffval2  8475  cnfcomlem  8479  cnfcom  8480  cnfcom3lem  8483  cnfcom3  8484  infxpenc  8724