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Theorem ufildom1 22534
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 5055 . 2 ( 𝐹 = ∅ → ( 𝐹 ≼ 1o ↔ ∅ ≼ 1o))
2 uffixsn 22533 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ∈ 𝐹)
3 intss1 4877 . . . . . . . . 9 ({𝑥} ∈ 𝐹 𝐹 ⊆ {𝑥})
42, 3syl 17 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝐹 ⊆ {𝑥})
5 simpr 488 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝑥 𝐹)
65snssd 4726 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ⊆ 𝐹)
74, 6eqssd 3970 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝐹 = {𝑥})
87ex 416 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥 𝐹 𝐹 = {𝑥}))
98eximdv 1919 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 𝐹 → ∃𝑥 𝐹 = {𝑥}))
10 n0 4293 . . . . 5 ( 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 𝐹)
11 en1 8572 . . . . 5 ( 𝐹 ≈ 1o ↔ ∃𝑥 𝐹 = {𝑥})
129, 10, 113imtr4g 299 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 ≠ ∅ → 𝐹 ≈ 1o))
1312imp 410 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 ≠ ∅) → 𝐹 ≈ 1o)
14 endom 8532 . . 3 ( 𝐹 ≈ 1o 𝐹 ≼ 1o)
1513, 14syl 17 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 ≠ ∅) → 𝐹 ≼ 1o)
16 1on 8105 . . 3 1o ∈ On
17 0domg 8641 . . 3 (1o ∈ On → ∅ ≼ 1o)
1816, 17mp1i 13 . 2 (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1o)
191, 15, 18pm2.61ne 3099 1 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ≼ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2115  wne 3014  wss 3919  c0 4276  {csn 4550   cint 4862   class class class wbr 5052  Oncon0 6178  cfv 6343  1oc1o 8091  cen 8502  cdom 8503  UFilcufil 22507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1o 8098  df-en 8506  df-dom 8507  df-fbas 20542  df-fg 20543  df-fil 22454  df-ufil 22509
This theorem is referenced by: (None)
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