Theorem oei0 6285

Description: Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oei0	⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)

Proof of Theorem oei0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elon 4252	. . 3 ⊢ ∅ ∈ On
2		oeiv 6282	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ↑o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
3	1, 2	mpan2 419	. 2 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
4		1on 6250	. . 3 ⊢ 1o ∈ On
5		rdg0g 6215	. . 3 ⊢ (1o ∈ On → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o)
6	4, 5	ax-mp 7	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o
7	3, 6	syl6eq 2148	1 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)

Syntax hints:   → wi 4   = wceq 1299   ∈ wcel 1448  Vcvv 2641  ∅c0 3310   ↦ cmpt 3929  Oncon0 4223  ‘cfv 5059  (class class class)co 5706  reccrdg 6196  1_oc1o 6236   ·_o comu 6241   ↑_o coei 6242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-oadd 6247  df-omul 6248  df-oexpi 6249

This theorem is referenced by: (None)