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Theorem ufildom1 23421
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 5150 . 2 ( 𝐹 = ∅ → ( 𝐹 ≼ 1o ↔ ∅ ≼ 1o))
2 uffixsn 23420 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ∈ 𝐹)
3 intss1 4966 . . . . . . . . 9 ({𝑥} ∈ 𝐹 𝐹 ⊆ {𝑥})
42, 3syl 17 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝐹 ⊆ {𝑥})
5 simpr 485 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝑥 𝐹)
65snssd 4811 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → {𝑥} ⊆ 𝐹)
74, 6eqssd 3998 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 𝐹) → 𝐹 = {𝑥})
87ex 413 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥 𝐹 𝐹 = {𝑥}))
98eximdv 1920 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 𝐹 → ∃𝑥 𝐹 = {𝑥}))
10 n0 4345 . . . . 5 ( 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 𝐹)
11 en1 9017 . . . . 5 ( 𝐹 ≈ 1o ↔ ∃𝑥 𝐹 = {𝑥})
129, 10, 113imtr4g 295 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 ≠ ∅ → 𝐹 ≈ 1o))
1312imp 407 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 ≠ ∅) → 𝐹 ≈ 1o)
14 endom 8971 . . 3 ( 𝐹 ≈ 1o 𝐹 ≼ 1o)
1513, 14syl 17 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 ≠ ∅) → 𝐹 ≼ 1o)
16 1on 8474 . . 3 1o ∈ On
17 0domg 9096 . . 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 396   = wceq 1541  wex 1781  wcel 2106  wne 2940  wss 3947  c0 4321  {csn 4627   cint 4949   class class class wbr 5147  Oncon0 6361  cfv 6540  1oc1o 8455  cen 8932  cdom 8933  UFilcufil 23394
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1o 8462  df-en 8936  df-dom 8937  df-fbas 20933  df-fg 20934  df-fil 23341  df-ufil 23396
This theorem is referenced by: (None)
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