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Theorem fi0 8902
 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 5170 . . 3 ∅ ∈ V
2 fival 8894 . . 3 (∅ ∈ V → (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥})
31, 2ax-mp 5 . 2 (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥}
4 vprc 5178 . . . 4 ¬ V ∈ V
5 id 22 . . . . . . 7 (𝑦 = 𝑥𝑦 = 𝑥)
6 elinel1 4096 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅)
7 elpwi 4496 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅)
8 ss0 4288 . . . . . . . . . 10 (𝑥 ⊆ ∅ → 𝑥 = ∅)
96, 7, 83syl 18 . . . . . . . . 9 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
109inteqd 4836 . . . . . . . 8 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
11 int0 4845 . . . . . . . 8 ∅ = V
1210, 11eqtrdi 2810 . . . . . . 7 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = V)
135, 12sylan9eqr 2816 . . . . . 6 ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = V)
1413rexlimiva 3203 . . . . 5 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥𝑦 = V)
15 vex 3411 . . . . 5 𝑦 ∈ V
1614, 15eqeltrrdi 2860 . . . 4 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥 → V ∈ V)
174, 16mto 200 . . 3 ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥
1817abf 4292 . 2 {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥} = ∅
193, 18eqtri 2782 1 (fi‘∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1539   ∈ wcel 2112  {cab 2736  ∃wrex 3069  Vcvv 3407   ∩ cin 3853   ⊆ wss 3854  ∅c0 4221  𝒫 cpw 4487  ∩ cint 4831  ‘cfv 6328  Fincfn 8520  ficfi 8892 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-sbc 3694  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-int 4832  df-br 5026  df-opab 5088  df-mpt 5106  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-iota 6287  df-fun 6330  df-fv 6336  df-fi 8893 This theorem is referenced by:  fieq0  8903  firest  16749  restbas  21843
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