Theorem oecl 3140

Description: Closure law for ordinal exponentiation.

Assertion

Ref	Expression
oecl	⊢ ((A ∈ On ∧ B ∈ On) → (A ↑o B) ∈ On)

Proof of Theorem oecl

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . . . 6 ⊢ (A = ∅ → (A ↑o B) = (∅ ↑o B))
2	1	eleq1d 1155	. . . . 5 ⊢ (A = ∅ → ((A ↑o B) ∈ On ↔ (∅ ↑o B) ∈ On))
3		oe0m0 3128	. . . . . . . . . 10 ⊢ (∅ ↑o ∅) = 1o
4		1o 3109	. . . . . . . . . 10 ⊢ 1o ∈ On
5	3, 4	eqeltr 1159	. . . . . . . . 9 ⊢ (∅ ↑o ∅) ∈ On
6		opreq2 3007	. . . . . . . . . 10 ⊢ (B = ∅ → (∅ ↑o B) = (∅ ↑o ∅))
7	6	eleq1d 1155	. . . . . . . . 9 ⊢ (B = ∅ → ((∅ ↑o B) ∈ On ↔ (∅ ↑o ∅) ∈ On))
8	5, 7	mpbiri 169	. . . . . . . 8 ⊢ (B = ∅ → (∅ ↑o B) ∈ On)
9	8	adantl 305	. . . . . . 7 ⊢ ((B ∈ On ∧ B = ∅) → (∅ ↑o B) ∈ On)
10		0elon 2277	. . . . . . . . 9 ⊢ ∅ ∈ On
11		oe0m1 3129	. . . . . . . . . 10 ⊢ ((B ∈ On ∧ ∅ ∈ B) → (∅ ↑o B) = ∅)
12	11	eleq1d 1155	. . . . . . . . 9 ⊢ ((B ∈ On ∧ ∅ ∈ B) → ((∅ ↑o B) ∈ On ↔ ∅ ∈ On))
13	10, 12	mpbiri 169	. . . . . . . 8 ⊢ ((B ∈ On ∧ ∅ ∈ B) → (∅ ↑o B) ∈ On)
14	13	adantll 309	. . . . . . 7 ⊢ (((B ∈ On ∧ B ∈ On) ∧ ∅ ∈ B) → (∅ ↑o B) ∈ On)
15	9, 14	oe0lem 3121	. . . . . 6 ⊢ ((B ∈ On ∧ B ∈ On) → (∅ ↑o B) ∈ On)
16	15	anidms 332	. . . . 5 ⊢ (B ∈ On → (∅ ↑o B) ∈ On)
17	2, 16	syl5bir 184	. . . 4 ⊢ (A = ∅ → (B ∈ On → (A ↑o B) ∈ On))
18	17	com12 13	. . 3 ⊢ (B ∈ On → (A = ∅ → (A ↑o B) ∈ On))
19	18	imp 277	. 2 ⊢ ((B ∈ On ∧ A = ∅) → (A ↑o B) ∈ On)
20		opreq2 3007	. . . . . . 7 ⊢ (x = ∅ → (A ↑o x) = (A ↑o ∅))
21	20	eleq1d 1155	. . . . . 6 ⊢ (x = ∅ → ((A ↑o x) ∈ On ↔ (A ↑o ∅) ∈ On))
22		opreq2 3007	. . . . . . 7 ⊢ (x = y → (A ↑o x) = (A ↑o y))
23	22	eleq1d 1155	. . . . . 6 ⊢ (x = y → ((A ↑o x) ∈ On ↔ (A ↑o y) ∈ On))
24		opreq2 3007	. . . . . . 7 ⊢ (x = suc y → (A ↑o x) = (A ↑o suc y))
25	24	eleq1d 1155	. . . . . 6 ⊢ (x = suc y → ((A ↑o x) ∈ On ↔ (A ↑o suc y) ∈ On))
26		opreq2 3007	. . . . . . 7 ⊢ (x = B → (A ↑o x) = (A ↑o B))
27	26	eleq1d 1155	. . . . . 6 ⊢ (x = B → ((A ↑o x) ∈ On ↔ (A ↑o B) ∈ On))
28		oe0 3130	. . . . . . . . 9 ⊢ (A ∈ On → (A ↑o ∅) = 1o)
29	28	eleq1d 1155	. . . . . . . 8 ⊢ (A ∈ On → ((A ↑o ∅) ∈ On ↔ 1o ∈ On))
30	4, 29	mpbiri 169	. . . . . . 7 ⊢ (A ∈ On → (A ↑o ∅) ∈ On)
31	30	adantr 306	. . . . . 6 ⊢ ((A ∈ On ∧ ∅ ∈ A) → (A ↑o ∅) ∈ On)
32		oesuc 3134	. . . . . . . . . . . . 13 ⊢ ((A ∈ On ∧ y ∈ On) → (A ↑o suc y) = ((A ↑o y) ·o A))
33	32	eleq1d 1155	. . . . . . . . . . . 12 ⊢ ((A ∈ On ∧ y ∈ On) → ((A ↑o suc y) ∈ On ↔ ((A ↑o y) ·o A) ∈ On))
34		omcl 3139	. . . . . . . . . . . 12 ⊢ (((A ↑o y) ∈ On ∧ A ∈ On) → ((A ↑o y) ·o A) ∈ On)
35	33, 34	syl5bir 184	. . . . . . . . . . 11 ⊢ ((A ∈ On ∧ y ∈ On) → (((A ↑o y) ∈ On ∧ A ∈ On) → (A ↑o suc y) ∈ On))
36	35	exp4b 296	. . . . . . . . . 10 ⊢ (A ∈ On → (y ∈ On → ((A ↑o y) ∈ On → (A ∈ On → (A ↑o suc y) ∈ On))))
37	36	com24 37	. . . . . . . . 9 ⊢ (A ∈ On → (A ∈ On → ((A ↑o y) ∈ On → (y ∈ On → (A ↑o suc y) ∈ On))))
38	37	pm2.43i 58	. . . . . . . 8 ⊢ (A ∈ On → ((A ↑o y) ∈ On → (y ∈ On → (A ↑o suc y) ∈ On)))
39	38	com3r 35	. . . . . . 7 ⊢ (y ∈ On → (A ∈ On → ((A ↑o y) ∈ On → (A ↑o suc y) ∈ On)))
40	39	adantrd 308	. . . . . 6 ⊢ (y ∈ On → ((A ∈ On ∧ ∅ ∈ A) → ((A ↑o y) ∈ On → (A ↑o suc y) ∈ On)))
41		visset 1350	. . . . . . . . . . . 12 ⊢ x ∈ V
42		oelim 3137	. . . . . . . . . . . 12 ⊢ (((A ∈ On ∧ (x ∈ V ∧ Lim x)) ∧ ∅ ∈ A) → (A ↑o x) = ∪y ∈ x (A ↑o y))
43	41, 42	mpan121 533	. . . . . . . . . . 11 ⊢ (((A ∈ On ∧ Lim x) ∧ ∅ ∈ A) → (A ↑o x) = ∪y ∈ x (A ↑o y))
44	43	anasss 337	. . . . . . . . . 10 ⊢ ((A ∈ On ∧ (Lim x ∧ ∅ ∈ A)) → (A ↑o x) = ∪y ∈ x (A ↑o y))
45	44	an1s 372	. . . . . . . . 9 ⊢ ((Lim x ∧ (A ∈ On ∧ ∅ ∈ A)) → (A ↑o x) = ∪y ∈ x (A ↑o y))
46	45	eleq1d 1155	. . . . . . . 8 ⊢ ((Lim x ∧ (A ∈ On ∧ ∅ ∈ A)) → ((A ↑o x) ∈ On ↔ ∪y ∈ x (A ↑o y) ∈ On))
47		oprex 3018	. . . . . . . . 9 ⊢ (A ↑o y) ∈ V
48	41, 47	iunon 2947	. . . . . . . 8 ⊢ (∀y ∈ x (A ↑o y) ∈ On → ∪y ∈ x (A ↑o y) ∈ On)
49	46, 48	syl5bir 184	. . . . . . 7 ⊢ ((Lim x ∧ (A ∈ On ∧ ∅ ∈ A)) → (∀y ∈ x (A ↑o y) ∈ On → (A ↑o x) ∈ On))
50	49	exp 291	. . . . . 6 ⊢ (Lim x → ((A ∈ On ∧ ∅ ∈ A) → (∀y ∈ x (A ↑o y) ∈ On → (A ↑o x) ∈ On)))
51	21, 23, 25, 27, 31, 40, 50	tfinds3 2406	. . . . 5 ⊢ (B ∈ On → ((A ∈ On ∧ ∅ ∈ A) → (A ↑o B) ∈ On))
52	51	exp3a 292	. . . 4 ⊢ (B ∈ On → (A ∈ On → (∅ ∈ A → (A ↑o B) ∈ On)))
53	52	com12 13	. . 3 ⊢ (A ∈ On → (B ∈ On → (∅ ∈ A → (A ↑o B) ∈ On)))
54	53	imp31 280	. 2 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → (A ↑o B) ∈ On)
55	19, 54	oe0lem 3121	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A ↑o B) ∈ On)

Syntax hints:   → wi 2   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348  ∅c0 1707  ∪ciun 1994  Oncon0 2199  Lim wlim 2200  suc csuc 2201  (class class class)co 3001  1_oc1o 3099   ·_o comu 3102   ↑_o coe 3103

This theorem is referenced by:  oen0 3165

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-oexp 3108

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