MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fi0 Structured version   Visualization version   GIF version

Theorem fi0 9415
Description: The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
fi0 (fi‘∅) = ∅

Proof of Theorem fi0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 5308 . . 3 ∅ ∈ V
2 fival 9407 . . 3 (∅ ∈ V → (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥})
31, 2ax-mp 5 . 2 (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥}
4 vprc 5316 . . . 4 ¬ V ∈ V
5 id 22 . . . . . . 7 (𝑦 = 𝑥𝑦 = 𝑥)
6 elinel1 4196 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅)
7 elpwi 4610 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅)
8 ss0 4399 . . . . . . . . . 10 (𝑥 ⊆ ∅ → 𝑥 = ∅)
96, 7, 83syl 18 . . . . . . . . 9 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
109inteqd 4956 . . . . . . . 8 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
11 int0 4967 . . . . . . . 8 ∅ = V
1210, 11eqtrdi 2789 . . . . . . 7 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = V)
135, 12sylan9eqr 2795 . . . . . 6 ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = V)
1413rexlimiva 3148 . . . . 5 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥𝑦 = V)
15 vex 3479 . . . . 5 𝑦 ∈ V
1614, 15eqeltrrdi 2843 . . . 4 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥 → V ∈ V)
174, 16mto 196 . . 3 ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥
1817abf 4403 . 2 {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥} = ∅
193, 18eqtri 2761 1 (fi‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  Vcvv 3475  cin 3948  wss 3949  c0 4323  𝒫 cpw 4603   cint 4951  cfv 6544  Fincfn 8939  ficfi 9405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-fi 9406
This theorem is referenced by:  fieq0  9416  firest  17378  restbas  22662
  Copyright terms: Public domain W3C validator