Theorem oecl 8506

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 7404	. . . . . . . 8 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2		oe0m0 8489	. . . . . . . . 9 ⊢ (∅ ↑o ∅) = 1o
3		1on 8450	. . . . . . . . 9 ⊢ 1o ∈ On
4	2, 3	eqeltri 2858	. . . . . . . 8 ⊢ (∅ ↑o ∅) ∈ On
5	1, 4	eqeltrdi 2870	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
6	5	adantl 485	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
7		oe0m1 8490	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
8	7	biimpa 480	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
9		0elon 6401	. . . . . . . 8 ⊢ ∅ ∈ On
10	8, 9	eqeltrdi 2870	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
11	10	adantll 724	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
12	6, 11	oe0lem 8482	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
13	12	anidms 574	. . . 4 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
14		oveq1 7403	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
15	14	eleq1d 2847	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ∈ On ↔ (∅ ↑o 𝐵) ∈ On))
16	13, 15	imbitrrid 248	. . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ On → (𝐴 ↑o 𝐵) ∈ On))
17	16	impcom 411	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) ∈ On)
18		oveq2 7404	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
19	18	eleq1d 2847	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o ∅) ∈ On))
20		oveq2 7404	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
21	20	eleq1d 2847	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o 𝑦) ∈ On))
22		oveq2 7404	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
23	22	eleq1d 2847	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o suc 𝑦) ∈ On))
24		oveq2 7404	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
25	24	eleq1d 2847	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑o 𝑥) ∈ On ↔ (𝐴 ↑o 𝐵) ∈ On))
26		oe0 8491	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
27	26, 3	eqeltrdi 2870	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) ∈ On)
28	27	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) ∈ On)
29		omcl 8505	. . . . . . . . . . 11 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On)
30	29	expcom 417	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
31	30	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ On))
32		oesuc 8496	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
33	32	eleq1d 2847	. . . . . . . . 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 417	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
36	35	adantrd 495	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o suc 𝑦) ∈ On)))
37		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
38		iunon 8310	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On)
39	37, 38	mpan 700	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On)
40		oelim 8503	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
41	37, 40	mpanlr1 716	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
42	41	anasss 470	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43	42	an12s 659	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
44	43	eleq1d 2847	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ((𝐴 ↑o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On))
45	39, 44	imbitrrid 248	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ∈ On → (𝐴 ↑o 𝑥) ∈ On))
46	45	ex 416	. . . . . 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 7845	. . . . 5 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On))
48	47	expd 419	. . . 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 421	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) ∈ On)
51	17, 50	oe0lem 8482	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  Vcvv 3454  ∅c0 4285  ∪ ciun 4949  Oncon0 6346  Lim wlim 6347  suc csuc 6348  (class class class)co 7396  1_oc1o 8430   ·_o comu 8435   ↑_o coe 8436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442  df-oexp 8443

This theorem is referenced by:  oen0  8556  oeordi  8557  oeord  8558  oecan  8559  oeword  8560  oewordri  8562  oeworde  8563  oeordsuc  8564  oeoalem  8566  oeoa  8567  oeoelem  8568  oeoe  8569  oelimcl  8570  oeeulem  8571  oeeui  8572  oaabs2  8619  omabs  8621  cantnfle  9626  cantnflt  9627  cantnfp1  9636  cantnflem1d  9643  cantnflem1  9644  cantnflem2  9645  cantnflem3  9646  cantnflem4  9647  cantnf  9648  oemapwe  9649  cantnffval2  9650  cnfcomlem  9654  cnfcom  9655  cnfcom3lem  9658  cnfcom3  9659  infxpenc  9974  onexoegt  43821  oaomoecl  43855  oenassex  43895  cantnftermord  43897  cantnfresb  43901  oacl2g  43907  omabs2  43909  omcl2  43910  ofoaf  43932  ofoafo  43933