Theorem oe0m 8444

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 6370	. . 3 ⊢ ∅ ∈ On
2		oev 8440	. . 3 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
3	1, 2	mpan 691	. 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
4		eqid 2737	. . 3 ⊢ ∅ = ∅
5	4	iftruei 4474	. 2 ⊢ if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)) = (1o ∖ 𝐴)
6	3, 5	eqtrdi 2788	1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∖ cdif 3887  ∅c0 4274  ifcif 4467   ↦ cmpt 5167  Oncon0 6315  ‘cfv 6490  (class class class)co 7358  reccrdg 8339  1_oc1o 8389   ·_o comu 8394   ↑_o coe 8395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-oexp 8402

This theorem is referenced by:  oe0m0  8446  oe0m1  8447  cantnflem2  9600  oe0rif  43716