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Theorem ufildom1 23950
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 5151 . 2 ( 𝐹 = ∅ → ( 𝐹 ≼ 1o ↔ ∅ ≼ 1o))
2 uffixsn 23949 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ∈ 𝐹)
3 intss1 4968 . . . . . . . . 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 4814 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ⊆ 𝐹)
74, 6eqssd 4013 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝐹 = {𝑥})
87ex 412 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥 𝐹 𝐹 = {𝑥}))
98eximdv 1915 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 𝐹 → ∃𝑥 𝐹 = {𝑥}))
10 n0 4359 . . . . 5 ( 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 𝐹)
11 en1 9063 . . . . 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 9018 . . 3 ( 𝐹 ≈ 1o 𝐹 ≼ 1o)
1513, 14syl 17 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 ≠ ∅) → 𝐹 ≼ 1o)
16 1on 8517 . . 3 1o ∈ On
17 0domg 9139 . . 3 (1o ∈ On → ∅ ≼ 1o)
1816, 17mp1i 13 . 2 (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1o)
191, 15, 18pm2.61ne 3025 1 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ≼ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  wss 3963  c0 4339  {csn 4631   cint 4951   class class class wbr 5148  Oncon0 6386  cfv 6563  1oc1o 8498  cen 8981  cdom 8982  UFilcufil 23923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-en 8985  df-dom 8986  df-fbas 21379  df-fg 21380  df-fil 23870  df-ufil 23925
This theorem is referenced by: (None)
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