Theorem oe0rif 43298

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 8556	. 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))
2		nel02 4339	. . . . . 6 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
3	2	iffalsed 4536	. . . . 5 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = 1o)
4		difeq2 4120	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
5		dif0 4378	. . . . . 6 ⊢ (1o ∖ ∅) = 1o
6	4, 5	eqtrdi 2793	. . . . 5 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
7	3, 6	eqtr4d 2780	. . . 4 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
8	7	adantl 481	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
9		iftrue 4531	. . . . 5 ⊢ (∅ ∈ 𝐴 → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
10	9	adantl 481	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
11		eloni 6394	. . . . . . 7 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordgt0ge1 8531	. . . . . . 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 4366	. . . . 5 ⊢ (1o ⊆ 𝐴 ↔ (1o ∖ 𝐴) = ∅)
16	14, 15	sylib 218	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (1o ∖ 𝐴) = ∅)
17	10, 16	eqtr4d 2780	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
18		on0eqel 6508	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19	8, 17, 18	mpjaodan 961	. 2 ⊢ (𝐴 ∈ On → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
20	1, 19	eqtr4d 2780	1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ ∈ 𝐴, ∅, 1o))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  ifcif 4525  Ord word 6383  Oncon0 6384  (class class class)co 7431  1_oc1o 8499   ↑_o coe 8505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oexp 8512

This theorem is referenced by: (None)