Theorem oe0m 8310

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 6304	. . 3 ⊢ ∅ ∈ On
2		oev 8306	. . 3 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
3	1, 2	mpan 686	. 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)))
4		eqid 2738	. . 3 ⊢ ∅ = ∅
5	4	iftruei 4463	. 2 ⊢ if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)) = (1o ∖ 𝐴)
6	3, 5	eqtrdi 2795	1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∖ cdif 3880  ∅c0 4253  ifcif 4456   ↦ cmpt 5153  Oncon0 6251  ‘cfv 6418  (class class class)co 7255  reccrdg 8211  1_oc1o 8260   ·_o comu 8265   ↑_o coe 8266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oexp 8273

This theorem is referenced by:  oe0m0  8312  oe0m1  8313  cantnflem2  9378