Theorem finxp0 37897

Description: The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.)

Assertion

Ref	Expression
finxp0	⊢ (𝑈↑↑∅) = ∅

Proof of Theorem finxp0

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

Step	Hyp	Ref	Expression
1		0ex 5262	. . . . 5 ⊢ ∅ ∈ V
2		vex 3461	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5447	. . . 4 ⊢ ⟨∅, 𝑦⟩ ≠ ∅
4	3	nesymi 3017	. . 3 ⊢ ¬ ∅ = ⟨∅, 𝑦⟩
5		peano1 7873	. . . . 5 ⊢ ∅ ∈ ω
6		df-finxp 37890	. . . . . 6 ⊢ (𝑈↑↑∅) = {𝑦 ∣ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅))}
7	6	eqabri 2907	. . . . 5 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅)))
8	5, 7	mpbiran 721	. . . 4 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅))
9		opex 5436	. . . . . 6 ⊢ ⟨∅, 𝑦⟩ ∈ V
10	9	rdg0 8396	. . . . 5 ⊢ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅) = ⟨∅, 𝑦⟩
11	10	eqeq2i 2778	. . . 4 ⊢ (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅) ↔ ∅ = ⟨∅, 𝑦⟩)
12	8, 11	bitri 278	. . 3 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = ⟨∅, 𝑦⟩)
13	4, 12	mtbir 326	. 2 ⊢ ¬ 𝑦 ∈ (𝑈↑↑∅)
14	13	nel0 4310	1 ⊢ (𝑈↑↑∅) = ∅

Syntax hints:   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ∅c0 4288  ifcif 4483  ⟨cop 4591  ∪ cuni 4868   × cxp 5650  ‘cfv 6525   ∈ cmpo 7402  ωcom 7850  1^st c1st 7972  reccrdg 8384  1_oc1o 8434  ↑↑cfinxp 37889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-finxp 37890

This theorem is referenced by:  finxp00  37908