Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  finxp0 Structured version   Visualization version   GIF version

Theorem finxp0 37098
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 5312 . . . . 5 ∅ ∈ V
2 vex 3466 . . . . 5 𝑦 ∈ V
31, 2opnzi 5480 . . . 4 ⟨∅, 𝑦⟩ ≠ ∅
43nesymi 2988 . . 3 ¬ ∅ = ⟨∅, 𝑦
5 peano1 7900 . . . . 5 ∅ ∈ ω
6 df-finxp 37091 . . . . . 6 (𝑈↑↑∅) = {𝑦 ∣ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅))}
76eqabri 2870 . . . . 5 (𝑦 ∈ (𝑈↑↑∅) ↔ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅)))
85, 7mpbiran 707 . . . 4 (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅))
9 opex 5470 . . . . . 6 ⟨∅, 𝑦⟩ ∈ V
109rdg0 8451 . . . . 5 (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅) = ⟨∅, 𝑦
1110eqeq2i 2739 . . . 4 (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨∅, 𝑦⟩)‘∅) ↔ ∅ = ⟨∅, 𝑦⟩)
128, 11bitri 274 . . 3 (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = ⟨∅, 𝑦⟩)
134, 12mtbir 322 . 2 ¬ 𝑦 ∈ (𝑈↑↑∅)
1413nel0 4353 1 (𝑈↑↑∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  c0 4325  ifcif 4533  cop 4639   cuni 4913   × cxp 5680  cfv 6554  cmpo 7426  ωcom 7876  1st c1st 8001  reccrdg 8439  1oc1o 8489  ↑↑cfinxp 37090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-finxp 37091
This theorem is referenced by:  finxp00  37109
  Copyright terms: Public domain W3C validator