Theorem oe0rif 43995

Description: Ordinal zero raised to any nonzero 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 8504	. 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))
2		nel02 4293	. . . . . 6 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
3	2	iffalsed 4499	. . . . 5 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = 1o)
4		difeq2 4076	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
5		dif0 4335	. . . . . 6 ⊢ (1o ∖ ∅) = 1o
6	4, 5	eqtrdi 2814	. . . . 5 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
7	3, 6	eqtr4d 2801	. . . 4 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
8	7	adantl 486	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
9		iftrue 4494	. . . . 5 ⊢ (∅ ∈ 𝐴 → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
10	9	adantl 486	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
11		eloni 6372	. . . . . . 7 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordgt0ge1 8479	. . . . . . 7 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
13	11, 12	syl 18	. . . . . 6 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
14	13	biimpa 481	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → 1o ⊆ 𝐴)
15		ssdif0 4322	. . . . 5 ⊢ (1o ⊆ 𝐴 ↔ (1o ∖ 𝐴) = ∅)
16	14, 15	sylib 221	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (1o ∖ 𝐴) = ∅)
17	10, 16	eqtr4d 2801	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
18		on0eqel 6488	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19	8, 17, 18	mpjaodan 973	. 2 ⊢ (𝐴 ∈ On → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
20	1, 19	eqtr4d 2801	1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ ∈ 𝐴, ∅, 1o))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287  ifcif 4488  Ord word 6361  Oncon0 6362  (class class class)co 7412  1_oc1o 8447   ↑_o coe 8453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oexp 8460

This theorem is referenced by: (None)