Theorem finxp0 37760

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 5236	. . . . 5 ⊢ ∅ ∈ V
2		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5421	. . . 4 ⊢ ⟨∅, 𝑦⟩ ≠ ∅
4	3	nesymi 2992	. . 3 ⊢ ¬ ∅ = ⟨∅, 𝑦⟩
5		peano1 7836	. . . . 5 ⊢ ∅ ∈ ω
6		df-finxp 37753	. . . . . 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 2882	. . . . 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 715	. . . 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 5410	. . . . . 6 ⊢ ⟨∅, 𝑦⟩ ∈ V
10	9	rdg0 8357	. . . . 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 2753	. . . 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 276	. . 3 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = ⟨∅, 𝑦⟩)
13	4, 12	mtbir 324	. 2 ⊢ ¬ 𝑦 ∈ (𝑈↑↑∅)
14	13	nel0 4289	1 ⊢ (𝑈↑↑∅) = ∅

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ∅c0 4268  ifcif 4461  ⟨cop 4568  ∪ cuni 4845   × cxp 5623  ‘cfv 6492   ∈ cmpo 7365  ωcom 7813  1^st c1st 7936  reccrdg 8345  1_oc1o 8395  ↑↑cfinxp 37752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-finxp 37753

This theorem is referenced by:  finxp00  37771