Theorem oe0rif 44045

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 8505	. 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))
2		nel02 4292	. . . . . 6 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
3	2	iffalsed 4500	. . . . 5 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = 1o)
4		difeq2 4075	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
5		dif0 4334	. . . . . 6 ⊢ (1o ∖ ∅) = 1o
6	4, 5	eqtrdi 2816	. . . . 5 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
7	3, 6	eqtr4d 2803	. . . 4 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
8	7	adantl 487	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
9		iftrue 4495	. . . . 5 ⊢ (∅ ∈ 𝐴 → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
10	9	adantl 487	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
11		eloni 6374	. . . . . . 7 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordgt0ge1 8480	. . . . . . 7 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
13	11, 12	syl 18	. . . . . 6 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
14	13	biimpa 482	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → 1o ⊆ 𝐴)
15		ssdif0 4321	. . . . 5 ⊢ (1o ⊆ 𝐴 ↔ (1o ∖ 𝐴) = ∅)
16	14, 15	sylib 221	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (1o ∖ 𝐴) = ∅)
17	10, 16	eqtr4d 2803	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
18		on0eqel 6490	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19	8, 17, 18	mpjaodan 973	. 2 ⊢ (𝐴 ∈ On → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
20	1, 19	eqtr4d 2803	1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ ∈ 𝐴, ∅, 1o))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4286  ifcif 4489  Ord word 6363  Oncon0 6364  (class class class)co 7416  1_oc1o 8448   ↑_o coe 8454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  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 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oexp 8461

This theorem is used by: (None)