Theorem oe0 7647

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
oe0	⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)

Proof of Theorem oe0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6697	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 ∅) = (∅ ↑𝑜 ∅))
2		oe0m0 7645	. . . . 5 ⊢ (∅ ↑𝑜 ∅) = 1𝑜
3	1, 2	syl6eq 2701	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 ∅) = 1𝑜)
4	3	adantl 481	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 ∅) = 1𝑜)
5		0elon 5816	. . . . . 6 ⊢ ∅ ∈ On
6		oevn0 7640	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅))
7	5, 6	mpanl2 717	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅))
8		1on 7612	. . . . . . 7 ⊢ 1𝑜 ∈ On
9	8	elexi 3244	. . . . . 6 ⊢ 1𝑜 ∈ V
10	9	rdg0 7562	. . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅) = 1𝑜
11	7, 10	syl6eq 2701	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = 1𝑜)
12	11	adantll 750	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = 1𝑜)
13	4, 12	oe0lem 7638	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑𝑜 ∅) = 1𝑜)
14	13	anidms 678	1 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948   ↦ cmpt 4762  Oncon0 5761  ‘cfv 5926  (class class class)co 6690  reccrdg 7550  1_𝑜c1o 7598   ·_𝑜 comu 7603   ↑_𝑜 coe 7604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oexp 7611

This theorem is referenced by:  oecl  7662  oe1  7669  oe1m  7670  oen0  7711  oewordri  7717  oeoalem  7721  oeoelem  7723  oeoe  7724  oeeulem  7726  nnecl  7738  oaabs2  7770  cantnff  8609