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Theorem ufli 23418
Description: Property of a set that satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
ufli ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
Distinct variable groups:   𝑓,𝐹   𝑓,𝑋

Proof of Theorem ufli
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 isufl 23417 . . 3 (𝑋 ∈ UFL → (𝑋 ∈ UFL ↔ ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓))
21ibi 267 . 2 (𝑋 ∈ UFL → ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓)
3 sseq1 4008 . . . 4 (𝑔 = 𝐹 → (𝑔𝑓𝐹𝑓))
43rexbidv 3179 . . 3 (𝑔 = 𝐹 → (∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓 ↔ ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓))
54rspccva 3612 . 2 ((∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
62, 5sylan 581 1 ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  wss 3949  cfv 6544  Filcfil 23349  UFilcufil 23403  UFLcufl 23404
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-ext 2704
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  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-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ufl 23406
This theorem is referenced by:  ssufl  23422  ufldom  23466  ufilcmp  23536
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