Theorem oe1m 3147

Description: Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67.

Assertion

Ref	Expression
oe1m	⊢ (A ∈ On → (1o ↑o A) = 1o)

Proof of Theorem oe1m

Step	Hyp	Ref	Expression
1		opreq2 3007	. . 3 ⊢ (x = ∅ → (1o ↑o x) = (1o ↑o ∅))
2	1	cleq1d 1109	. 2 ⊢ (x = ∅ → ((1o ↑o x) = 1o ↔ (1o ↑o ∅) = 1o))
3		opreq2 3007	. . 3 ⊢ (x = y → (1o ↑o x) = (1o ↑o y))
4	3	cleq1d 1109	. 2 ⊢ (x = y → ((1o ↑o x) = 1o ↔ (1o ↑o y) = 1o))
5		opreq2 3007	. . 3 ⊢ (x = suc y → (1o ↑o x) = (1o ↑o suc y))
6	5	cleq1d 1109	. 2 ⊢ (x = suc y → ((1o ↑o x) = 1o ↔ (1o ↑o suc y) = 1o))
7		opreq2 3007	. . 3 ⊢ (x = A → (1o ↑o x) = (1o ↑o A))
8	7	cleq1d 1109	. 2 ⊢ (x = A → ((1o ↑o x) = 1o ↔ (1o ↑o A) = 1o))
9		1o 3109	. . 3 ⊢ 1o ∈ On
10		oe0 3130	. . 3 ⊢ (1o ∈ On → (1o ↑o ∅) = 1o)
11	9, 10	ax-mp 6	. 2 ⊢ (1o ↑o ∅) = 1o
12		oesuc 3134	. . . . 5 ⊢ ((1o ∈ On ∧ y ∈ On) → (1o ↑o suc y) = ((1o ↑o y) ·o 1o))
13	9, 12	mpan 518	. . . 4 ⊢ (y ∈ On → (1o ↑o suc y) = ((1o ↑o y) ·o 1o))
14		opreq1 3006	. . . . 5 ⊢ ((1o ↑o y) = 1o → ((1o ↑o y) ·o 1o) = (1o ·o 1o))
15		om1 3144	. . . . . 6 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
16	9, 15	ax-mp 6	. . . . 5 ⊢ (1o ·o 1o) = 1o
17	14, 16	syl6eq 1140	. . . 4 ⊢ ((1o ↑o y) = 1o → ((1o ↑o y) ·o 1o) = 1o)
18	13, 17	sylan9eq 1144	. . 3 ⊢ ((y ∈ On ∧ (1o ↑o y) = 1o) → (1o ↑o suc y) = 1o)
19	18	exp 291	. 2 ⊢ (y ∈ On → ((1o ↑o y) = 1o → (1o ↑o suc y) = 1o))
20		visset 1350	. . . . . 6 ⊢ x ∈ V
21		0lt1o 3116	. . . . . . . 8 ⊢ ∅ ∈ 1o
22		oelim 3137	. . . . . . . 8 ⊢ (((1o ∈ On ∧ (x ∈ V ∧ Lim x)) ∧ ∅ ∈ 1o) → (1o ↑o x) = ∪y ∈ x (1o ↑o y))
23	21, 22	mpan2 519	. . . . . . 7 ⊢ ((1o ∈ On ∧ (x ∈ V ∧ Lim x)) → (1o ↑o x) = ∪y ∈ x (1o ↑o y))
24	9, 23	mpan 518	. . . . . 6 ⊢ ((x ∈ V ∧ Lim x) → (1o ↑o x) = ∪y ∈ x (1o ↑o y))
25	20, 24	mpan 518	. . . . 5 ⊢ (Lim x → (1o ↑o x) = ∪y ∈ x (1o ↑o y))
26	25	cleq1d 1109	. . . 4 ⊢ (Lim x → ((1o ↑o x) = 1o ↔ ∪y ∈ x (1o ↑o y) = 1o))
27		0ellim 2285	. . . . . 6 ⊢ (Lim x → ∅ ∈ x)
28		n0i 1712	. . . . . 6 ⊢ (∅ ∈ x → ¬ x = ∅)
29		iunconst 2000	. . . . . 6 ⊢ (¬ x = ∅ → ∪y ∈ x 1o = 1o)
30	27, 28, 29	3syl 21	. . . . 5 ⊢ (Lim x → ∪y ∈ x 1o = 1o)
31	30	cleq2d 1112	. . . 4 ⊢ (Lim x → (∪y ∈ x (1o ↑o y) = ∪y ∈ x 1o ↔ ∪y ∈ x (1o ↑o y) = 1o))
32	26, 31	bitr4d 409	. . 3 ⊢ (Lim x → ((1o ↑o x) = 1o ↔ ∪y ∈ x (1o ↑o y) = ∪y ∈ x 1o))
33		iuneq2 2006	. . 3 ⊢ (∀y ∈ x (1o ↑o y) = 1o → ∪y ∈ x (1o ↑o y) = ∪y ∈ x 1o)
34	32, 33	syl5bir 184	. 2 ⊢ (Lim x → (∀y ∈ x (1o ↑o y) = 1o → (1o ↑o x) = 1o))
35	2, 4, 6, 8, 11, 19, 34	tfinds 2401	1 ⊢ (A ∈ On → (1o ↑o A) = 1o)

Syntax hints:  ¬ wn 1   → 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 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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