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Theorem ufli 21699
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 21698 . . 3 (𝑋 ∈ UFL → (𝑋 ∈ UFL ↔ ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓))
21ibi 256 . 2 (𝑋 ∈ UFL → ∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓)
3 sseq1 3618 . . . 4 (𝑔 = 𝐹 → (𝑔𝑓𝐹𝑓))
43rexbidv 3048 . . 3 (𝑔 = 𝐹 → (∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓 ↔ ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓))
54rspccva 3303 . 2 ((∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
62, 5sylan 488 1 ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  wrex 2910  wss 3567  cfv 5876  Filcfil 21630  UFilcufil 21684  UFLcufl 21685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ufl 21687
This theorem is referenced by:  ssufl  21703  ufldom  21747  ufilcmp  21817
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