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Theorem fi0 8367
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 4823 . . 3 ∅ ∈ V
2 fival 8359 . . 3 (∅ ∈ V → (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥})
31, 2ax-mp 5 . 2 (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥}
4 vprc 4829 . . . 4 ¬ V ∈ V
5 id 22 . . . . . . 7 (𝑦 = 𝑥𝑦 = 𝑥)
6 inss1 3866 . . . . . . . . . . 11 (𝒫 ∅ ∩ Fin) ⊆ 𝒫 ∅
76sseli 3632 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅)
8 elpwi 4201 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅)
9 ss0 4007 . . . . . . . . . 10 (𝑥 ⊆ ∅ → 𝑥 = ∅)
107, 8, 93syl 18 . . . . . . . . 9 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
1110inteqd 4512 . . . . . . . 8 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
12 int0 4522 . . . . . . . 8 ∅ = V
1311, 12syl6eq 2701 . . . . . . 7 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = V)
145, 13sylan9eqr 2707 . . . . . 6 ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = V)
1514rexlimiva 3057 . . . . 5 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥𝑦 = V)
16 vex 3234 . . . . 5 𝑦 ∈ V
1715, 16syl6eqelr 2739 . . . 4 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥 → V ∈ V)
184, 17mto 188 . . 3 ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥
1918abf 4011 . 2 {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥} = ∅
203, 19eqtri 2673 1 (fi‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  Vcvv 3231  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   cint 4507  cfv 5926  Fincfn 7997  ficfi 8357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-fi 8358
This theorem is referenced by:  fieq0  8368  firest  16140  restbas  21010
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