Theorem oe0rif 43258

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 8443	. 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴))
2		nel02 4292	. . . . . 6 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴)
3	2	iffalsed 4489	. . . . 5 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = 1o)
4		difeq2 4073	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
5		dif0 4331	. . . . . 6 ⊢ (1o ∖ ∅) = 1o
6	4, 5	eqtrdi 2780	. . . . 5 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
7	3, 6	eqtr4d 2767	. . . 4 ⊢ (𝐴 = ∅ → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
8	7	adantl 481	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
9		iftrue 4484	. . . . 5 ⊢ (∅ ∈ 𝐴 → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
10	9	adantl 481	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = ∅)
11		eloni 6321	. . . . . . 7 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordgt0ge1 8418	. . . . . . 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 4319	. . . . 5 ⊢ (1o ⊆ 𝐴 ↔ (1o ∖ 𝐴) = ∅)
16	14, 15	sylib 218	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (1o ∖ 𝐴) = ∅)
17	10, 16	eqtr4d 2767	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
18		on0eqel 6436	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19	8, 17, 18	mpjaodan 960	. 2 ⊢ (𝐴 ∈ On → if(∅ ∈ 𝐴, ∅, 1o) = (1o ∖ 𝐴))
20	1, 19	eqtr4d 2767	1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ ∈ 𝐴, ∅, 1o))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  ifcif 4478  Ord word 6310  Oncon0 6311  (class class class)co 7353  1_oc1o 8388   ↑_o coe 8394

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oexp 8401

This theorem is referenced by: (None)