Theorem oe0m 8513

Description: Value of zero raised to an ordinal. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oe0m	⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))

Proof of Theorem oe0m

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elon 6408	. . 3 ⊢ ∅ ∈ On
2		oev 8509	. . 3 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
3	1, 2	mpan 687	. 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
4		eqid 2724	. . 3 ⊢ ∅ = ∅
5	4	iftruei 4527	. 2 ⊢ if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)) = (1o ∖ 𝐴)
6	3, 5	eqtrdi 2780	1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∖ cdif 3937  ∅c0 4314  ifcif 4520   ↦ cmpt 5221  Oncon0 6354  ‘cfv 6533  (class class class)co 7401  reccrdg 8404  1_oc1o 8454   ·_o comu 8459   ↑_o coe 8460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oexp 8467

This theorem is referenced by:  oe0m0  8515  oe0m1  8516  cantnflem2  9681  oe0rif  42524