Theorem finxpnom 35099

Description: Cartesian exponentiation when the exponent is not a natural number defaults to the empty set. (Contributed by ML, 24-Oct-2020.)

Assertion

Ref	Expression
finxpnom	⊢ (¬ 𝑁 ∈ ω → (𝑈↑↑𝑁) = ∅)

Proof of Theorem finxpnom

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 487	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) → 𝑁 ∈ ω)
2	1	con3i 157	. . . 4 ⊢ (¬ 𝑁 ∈ ω → ¬ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)))
3		abid 2740	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} ↔ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)))
4	2, 3	sylnibr 333	. . 3 ⊢ (¬ 𝑁 ∈ ω → ¬ 𝑦 ∈ {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))})
5		df-finxp 35082	. . . 4 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
6	5	eleq2i 2844	. . 3 ⊢ (𝑦 ∈ (𝑈↑↑𝑁) ↔ 𝑦 ∈ {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))})
7	4, 6	sylnibr 333	. 2 ⊢ (¬ 𝑁 ∈ ω → ¬ 𝑦 ∈ (𝑈↑↑𝑁))
8	7	eq0rdv 4301	1 ⊢ (¬ 𝑁 ∈ ω → (𝑈↑↑𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {cab 2736  Vcvv 3410  ∅c0 4226  ifcif 4421  ⟨cop 4529  ∪ cuni 4799   × cxp 5523  ‘cfv 6336   ∈ cmpo 7153  ωcom 7580  1^st c1st 7692  reccrdg 8056  1_oc1o 8106  ↑↑cfinxp 35081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-dif 3862  df-nul 4227  df-finxp 35082

This theorem is referenced by:  finxp00  35100