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Theorem ufildom1 23934
Description: An ultrafilter is generated by at most one element (because free ultrafilters have no generators and fixed ultrafilters have exactly one). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
ufildom1 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ≼ 1o)

Proof of Theorem ufildom1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq1 5146 . 2 ( 𝐹 = ∅ → ( 𝐹 ≼ 1o ↔ ∅ ≼ 1o))
2 uffixsn 23933 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ∈ 𝐹)
3 intss1 4963 . . . . . . . . 9 ({𝑥} ∈ 𝐹 𝐹 ⊆ {𝑥})
42, 3syl 17 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝐹 ⊆ {𝑥})
5 simpr 484 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝑥 𝐹)
65snssd 4809 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ⊆ 𝐹)
74, 6eqssd 4001 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝐹 = {𝑥})
87ex 412 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥 𝐹 𝐹 = {𝑥}))
98eximdv 1917 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 𝐹 → ∃𝑥 𝐹 = {𝑥}))
10 n0 4353 . . . . 5 ( 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 𝐹)
11 en1 9064 . . . . 5 ( 𝐹 ≈ 1o ↔ ∃𝑥 𝐹 = {𝑥})
129, 10, 113imtr4g 296 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 ≠ ∅ → 𝐹 ≈ 1o))
1312imp 406 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 ≠ ∅) → 𝐹 ≈ 1o)
14 endom 9019 . . 3 ( 𝐹 ≈ 1o 𝐹 ≼ 1o)
1513, 14syl 17 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 ≠ ∅) → 𝐹 ≼ 1o)
16 1on 8518 . . 3 1o ∈ On
17 0domg 9140 . . 3 (1o ∈ On → ∅ ≼ 1o)
1816, 17mp1i 13 . 2 (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1o)
191, 15, 18pm2.61ne 3027 1 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ≼ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wss 3951  c0 4333  {csn 4626   cint 4946   class class class wbr 5143  Oncon0 6384  cfv 6561  1oc1o 8499  cen 8982  cdom 8983  UFilcufil 23907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8506  df-en 8986  df-dom 8987  df-fbas 21361  df-fg 21362  df-fil 23854  df-ufil 23909
This theorem is referenced by: (None)
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