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Theorem fi0 8270
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 4750 . . 3 ∅ ∈ V
2 fival 8262 . . 3 (∅ ∈ V → (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥})
31, 2ax-mp 5 . 2 (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥}
4 vprc 4756 . . . 4 ¬ V ∈ V
5 id 22 . . . . . . 7 (𝑦 = 𝑥𝑦 = 𝑥)
6 inss1 3811 . . . . . . . . . . 11 (𝒫 ∅ ∩ Fin) ⊆ 𝒫 ∅
76sseli 3579 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅)
8 elpwi 4140 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅)
9 ss0 3946 . . . . . . . . . 10 (𝑥 ⊆ ∅ → 𝑥 = ∅)
107, 8, 93syl 18 . . . . . . . . 9 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
1110inteqd 4445 . . . . . . . 8 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
12 int0 4455 . . . . . . . 8 ∅ = V
1311, 12syl6eq 2671 . . . . . . 7 (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = V)
145, 13sylan9eqr 2677 . . . . . 6 ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = V)
1514rexlimiva 3021 . . . . 5 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥𝑦 = V)
16 vex 3189 . . . . 5 𝑦 ∈ V
1715, 16syl6eqelr 2707 . . . 4 (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥 → V ∈ V)
184, 17mto 188 . . 3 ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥
1918abf 3950 . 2 {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = 𝑥} = ∅
203, 19eqtri 2643 1 (fi‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  Vcvv 3186  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130   cint 4440  cfv 5847  Fincfn 7899  ficfi 8260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-fi 8261
This theorem is referenced by:  fieq0  8271  firest  16014  restbas  20872
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