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Theorem fi0 9458
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 5313 . . 3 ∅ ∈ V
2 fival 9450 . . 3 (∅ ∈ V → (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥})
31, 2ax-mp 5 . 2 (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥}
4 vprc 5321 . . . 4 ¬ V ∈ V
5 id 22 . . . . . . 7 (𝑦 = 𝑥𝑦 = 𝑥)
6 elinel1 4211 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅)
7 elpwi 4612 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅)
8 ss0 4408 . . . . . . . . . 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 2791 . . . . . . 7 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = V)
135, 12sylan9eqr 2797 . . . . . 6 ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = V)
1413rexlimiva 3145 . . . . 5 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥𝑦 = V)
15 vex 3482 . . . . 5 𝑦 ∈ V
1614, 15eqeltrrdi 2848 . . . 4 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥 → V ∈ V)
174, 16mto 197 . . 3 ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥
1817abf 4412 . 2 {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥} = ∅
193, 18eqtri 2763 1 (fi‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cint 4951  cfv 6563  Fincfn 8984  ficfi 9448
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-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-fi 9449
This theorem is referenced by:  fieq0  9459  firest  17479  restbas  23182
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