Theorem oe0rif 43405

Description: Ordinal zero raised to any non-zero ordinal power is zero and zero to the zeroth power is one. Lemma 2.18 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.)

Assertion

Ref	Expression
oe0rif	⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ ∈ 𝐴, ∅, 1o))

Proof of Theorem oe0rif

Step	Hyp	Ref	Expression
1		oe0m 8441	. 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))
2		nel02 4288	. . . . . 6 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
3	2	iffalsed 4487	. . . . 5 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = 1o)
4		difeq2 4069	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
5		dif0 4327	. . . . . 6 ⊢ (1o ∖ ∅) = 1o
6	4, 5	eqtrdi 2784	. . . . 5 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
7	3, 6	eqtr4d 2771	. . . 4 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
8	7	adantl 481	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
9		iftrue 4482	. . . . 5 ⊢ (∅ ∈ 𝐴 → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
10	9	adantl 481	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
11		eloni 6323	. . . . . . 7 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordgt0ge1 8416	. . . . . . 7 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
13	11, 12	syl 17	. . . . . 6 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
14	13	biimpa 476	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → 1o ⊆ 𝐴)
15		ssdif0 4315	. . . . 5 ⊢ (1o ⊆ 𝐴 ↔ (1o ∖ 𝐴) = ∅)
16	14, 15	sylib 218	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (1o ∖ 𝐴) = ∅)
17	10, 16	eqtr4d 2771	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
18		on0eqel 6438	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19	8, 17, 18	mpjaodan 960	. 2 ⊢ (𝐴 ∈ On → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
20	1, 19	eqtr4d 2771	1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ ∈ 𝐴, ∅, 1o))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3895   ⊆ wss 3898  ∅c0 4282  ifcif 4476  Ord word 6312  Oncon0 6313  (class class class)co 7354  1_oc1o 8386   ↑_o coe 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-suc 6319  df-iota 6444  df-fun 6490  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-1o 8393  df-oexp 8399

This theorem is referenced by: (None)