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Theorem ufli 21937
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 21936 . . 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 3773 . . . 4 (𝑔 = 𝐹 → (𝑔𝑓𝐹𝑓))
43rexbidv 3199 . . 3 (𝑔 = 𝐹 → (∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓 ↔ ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓))
54rspccva 3457 . 2 ((∀𝑔 ∈ (Fil‘𝑋)∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
62, 5sylan 561 1 ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  wss 3721  cfv 6031  Filcfil 21868  UFilcufil 21922  UFLcufl 21923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ufl 21925
This theorem is referenced by:  ssufl  21941  ufldom  21985  ufilcmp  22055
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