Theorem oe0m 8487

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 6401	. . 3 ⊢ ∅ ∈ On
2		oev 8483	. . 3 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
3	1, 2	mpan 700	. 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
4		eqid 2763	. . 3 ⊢ ∅ = ∅
5	4	iftruei 4488	. 2 ⊢ if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)) = (1o ∖ 𝐴)
6	3, 5	eqtrdi 2814	1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902  ∅c0 4286  ifcif 4481   ↦ cmpt 5182  Oncon0 6346  ‘cfv 6521  (class class class)co 7396  reccrdg 8380  1_oc1o 8430   ·_o comu 8435   ↑_o coe 8436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-oexp 8443

This theorem is referenced by:  oe0m0  8489  oe0m1  8490  cantnflem2  9643  oe0rif  43867