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Theorem finxp0 33726
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.
StepHypRef Expression
1 0ex 4984 . . . . 5 ∅ ∈ V
2 vex 3388 . . . . 5 𝑦 ∈ V
31, 2opnzi 5133 . . . 4 ⟨∅, 𝑦⟩ ≠ ∅
43nesymi 3028 . . 3 ¬ ∅ = ⟨∅, 𝑦
5 peano1 7319 . . . . 5 ∅ ∈ ω
6 df-finxp 33719 . . . . . 6 (𝑈↑↑∅) = {𝑦 ∣ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅))}
76abeq2i 2912 . . . . 5 (𝑦 ∈ (𝑈↑↑∅) ↔ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅)))
85, 7mpbiran 701 . . . 4 (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅))
9 opex 5123 . . . . . 6 ⟨∅, 𝑦⟩ ∈ V
109rdg0 7756 . . . . 5 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅) = ⟨∅, 𝑦
1110eqeq2i 2811 . . . 4 (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅) ↔ ∅ = ⟨∅, 𝑦⟩)
128, 11bitri 267 . . 3 (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = ⟨∅, 𝑦⟩)
134, 12mtbir 315 . 2 ¬ 𝑦 ∈ (𝑈↑↑∅)
1413nel0 4132 1 (𝑈↑↑∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  c0 4115  ifcif 4277  cop 4374   cuni 4628   × cxp 5310  cfv 6101  cmpt2 6880  ωcom 7299  1st c1st 7399  reccrdg 7744  1𝑜c1o 7792  ↑↑cfinxp 33718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-finxp 33719
This theorem is referenced by:  finxp00  33737
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