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Theorem ufildom1 23785
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 5144 . 2 ( 𝐹 = ∅ → ( 𝐹 ≼ 1o ↔ ∅ ≼ 1o))
2 uffixsn 23784 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ∈ 𝐹)
3 intss1 4960 . . . . . . . . 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 4807 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ⊆ 𝐹)
74, 6eqssd 3994 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝐹 = {𝑥})
87ex 412 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥 𝐹 𝐹 = {𝑥}))
98eximdv 1912 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 𝐹 → ∃𝑥 𝐹 = {𝑥}))
10 n0 4341 . . . . 5 ( 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 𝐹)
11 en1 9023 . . . . 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 8977 . . 3 ( 𝐹 ≈ 1o 𝐹 ≼ 1o)
1513, 14syl 17 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 ≠ ∅) → 𝐹 ≼ 1o)
16 1on 8479 . . 3 1o ∈ On
17 0domg 9102 . . 3 (1o ∈ On → ∅ ≼ 1o)
1816, 17mp1i 13 . 2 (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1o)
191, 15, 18pm2.61ne 3021 1 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ≼ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  wne 2934  wss 3943  c0 4317  {csn 4623   cint 4943   class class class wbr 5141  Oncon0 6358  cfv 6537  1oc1o 8460  cen 8938  cdom 8939  UFilcufil 23758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1o 8467  df-en 8942  df-dom 8943  df-fbas 21237  df-fg 21238  df-fil 23705  df-ufil 23760
This theorem is referenced by: (None)
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