Theorem oe0 3130

Description: Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67.

Assertion

Ref	Expression
oe0	⊢ (A ∈ On → (A ↑o ∅) = 1o)

Proof of Theorem oe0

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . . 5 ⊢ (A = ∅ → (A ↑o ∅) = (∅ ↑o ∅))
2		oe0m0 3128	. . . . 5 ⊢ (∅ ↑o ∅) = 1o
3	1, 2	syl6eq 1140	. . . 4 ⊢ (A = ∅ → (A ↑o ∅) = 1o)
4	3	adantl 305	. . 3 ⊢ ((A ∈ On ∧ A = ∅) → (A ↑o ∅) = 1o)
5		0elon 2277	. . . . . 6 ⊢ ∅ ∈ On
6		oevn0 3123	. . . . . 6 ⊢ (((A ∈ On ∧ ∅ ∈ On) ∧ ∅ ∈ A) → (A ↑o ∅) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘∅))
7	5, 6	mpan12 530	. . . . 5 ⊢ ((A ∈ On ∧ ∅ ∈ A) → (A ↑o ∅) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘∅))
8		1o 3109	. . . . . . 7 ⊢ 1o ∈ On
9	8	elisseti 1355	. . . . . 6 ⊢ 1o ∈ V
10	9	rdgzer 2979	. . . . 5 ⊢ (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘∅) = 1o
11	7, 10	syl6eq 1140	. . . 4 ⊢ ((A ∈ On ∧ ∅ ∈ A) → (A ↑o ∅) = 1o)
12	11	adantll 309	. . 3 ⊢ (((A ∈ On ∧ A ∈ On) ∧ ∅ ∈ A) → (A ↑o ∅) = 1o)
13	4, 12	oe0lem 3121	. 2 ⊢ ((A ∈ On ∧ A ∈ On) → (A ↑o ∅) = 1o)
14	13	anidms 332	1 ⊢ (A ∈ On → (A ↑o ∅) = 1o)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707  {copab 2055  Oncon0 2199   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  1_oc1o 3099   ·_o comu 3102   ↑_o coe 3103

This theorem is referenced by:  oecl 3140  oe1 3146  oe1m 3147  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-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-oexp 3108

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