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Theorem finxp0 37374
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 5313 . . . . 5 ∅ ∈ V
2 vex 3482 . . . . 5 𝑦 ∈ V
31, 2opnzi 5485 . . . 4 ⟨∅, 𝑦⟩ ≠ ∅
43nesymi 2996 . . 3 ¬ ∅ = ⟨∅, 𝑦
5 peano1 7911 . . . . 5 ∅ ∈ ω
6 df-finxp 37367 . . . . . 6 (𝑈↑↑∅) = {𝑦 ∣ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅))}
76eqabri 2883 . . . . 5 (𝑦 ∈ (𝑈↑↑∅) ↔ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅)))
85, 7mpbiran 709 . . . 4 (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅))
9 opex 5475 . . . . . 6 ⟨∅, 𝑦⟩ ∈ V
109rdg0 8460 . . . . 5 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅) = ⟨∅, 𝑦
1110eqeq2i 2748 . . . 4 (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅) ↔ ∅ = ⟨∅, 𝑦⟩)
128, 11bitri 275 . . 3 (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = ⟨∅, 𝑦⟩)
134, 12mtbir 323 . 2 ¬ 𝑦 ∈ (𝑈↑↑∅)
1413nel0 4360 1 (𝑈↑↑∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  ifcif 4531  cop 4637   cuni 4912   × cxp 5687  cfv 6563  cmpo 7433  ωcom 7887  1st c1st 8011  reccrdg 8448  1oc1o 8498  ↑↑cfinxp 37366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-finxp 37367
This theorem is referenced by:  finxp00  37385
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