MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ufprim Structured version   Visualization version   GIF version

Theorem ufprim 23917
Description: An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)
Assertion
Ref Expression
ufprim ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐹𝐵𝐹) ↔ (𝐴𝐵) ∈ 𝐹))

Proof of Theorem ufprim
StepHypRef Expression
1 ufilfil 23912 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
213ad2ant1 1134 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → 𝐹 ∈ (Fil‘𝑋))
32adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐹 ∈ (Fil‘𝑋))
4 simpr 484 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐴𝐹)
5 unss 4190 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
65biimpi 216 . . . . . . 7 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
763adant1 1131 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
87adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ⊆ 𝑋)
9 ssun1 4178 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
109a1i 11 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝐴𝐵))
11 filss 23861 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝐴 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
123, 4, 8, 10, 11syl13anc 1374 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ∈ 𝐹)
1312ex 412 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐹 → (𝐴𝐵) ∈ 𝐹))
142adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐹 ∈ (Fil‘𝑋))
15 simpr 484 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐵𝐹)
167adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → (𝐴𝐵) ⊆ 𝑋)
17 ssun2 4179 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1817a1i 11 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐵 ⊆ (𝐴𝐵))
19 filss 23861 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐵𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝐵 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
2014, 15, 16, 18, 19syl13anc 1374 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → (𝐴𝐵) ∈ 𝐹)
2120ex 412 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐹 → (𝐴𝐵) ∈ 𝐹))
2213, 21jaod 860 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐹𝐵𝐹) → (𝐴𝐵) ∈ 𝐹))
23 ufilb 23914 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
24233adant3 1133 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
2524adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
2623ad2ant1 1134 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
27 difun2 4481 . . . . . . . . . . 11 ((𝐵𝐴) ∖ 𝐴) = (𝐵𝐴)
28 uncom 4158 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
2928difeq1i 4122 . . . . . . . . . . 11 ((𝐵𝐴) ∖ 𝐴) = ((𝐴𝐵) ∖ 𝐴)
3027, 29eqtr3i 2767 . . . . . . . . . 10 (𝐵𝐴) = ((𝐴𝐵) ∖ 𝐴)
3130ineq2i 4217 . . . . . . . . 9 (𝑋 ∩ (𝐵𝐴)) = (𝑋 ∩ ((𝐴𝐵) ∖ 𝐴))
32 indifcom 4283 . . . . . . . . 9 (𝐵 ∩ (𝑋𝐴)) = (𝑋 ∩ (𝐵𝐴))
33 indifcom 4283 . . . . . . . . 9 ((𝐴𝐵) ∩ (𝑋𝐴)) = (𝑋 ∩ ((𝐴𝐵) ∖ 𝐴))
3431, 32, 333eqtr4i 2775 . . . . . . . 8 (𝐵 ∩ (𝑋𝐴)) = ((𝐴𝐵) ∩ (𝑋𝐴))
35 filin 23862 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → ((𝐴𝐵) ∩ (𝑋𝐴)) ∈ 𝐹)
362, 35syl3an1 1164 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → ((𝐴𝐵) ∩ (𝑋𝐴)) ∈ 𝐹)
3734, 36eqeltrid 2845 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋𝐴)) ∈ 𝐹)
38 simp13 1206 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐵𝑋)
39 inss1 4237 . . . . . . . 8 (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵
4039a1i 11 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵)
41 filss 23861 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝐵 ∩ (𝑋𝐴)) ∈ 𝐹𝐵𝑋 ∧ (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵)) → 𝐵𝐹)
4226, 37, 38, 40, 41syl13anc 1374 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐵𝐹)
43423expia 1122 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → ((𝑋𝐴) ∈ 𝐹𝐵𝐹))
4425, 43sylbid 240 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (¬ 𝐴𝐹𝐵𝐹))
4544orrd 864 . . 3 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (𝐴𝐹𝐵𝐹))
4645ex 412 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐵) ∈ 𝐹 → (𝐴𝐹𝐵𝐹)))
4722, 46impbid 212 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐹𝐵𝐹) ↔ (𝐴𝐵) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087  wcel 2108  cdif 3948  cun 3949  cin 3950  wss 3951  cfv 6561  Filcfil 23853  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-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-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  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-iota 6514  df-fun 6563  df-fv 6569  df-fbas 21361  df-fil 23854  df-ufil 23909
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator