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Theorem ufprim 21760
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 21755 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
213ad2ant1 1102 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → 𝐹 ∈ (Fil‘𝑋))
32adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐹 ∈ (Fil‘𝑋))
4 simpr 476 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐴𝐹)
5 unss 3820 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
65biimpi 206 . . . . . . 7 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
763adant1 1099 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
87adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ⊆ 𝑋)
9 ssun1 3809 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
109a1i 11 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝐴𝐵))
11 filss 21704 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝐴 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
123, 4, 8, 10, 11syl13anc 1368 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ∈ 𝐹)
1312ex 449 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐹 → (𝐴𝐵) ∈ 𝐹))
142adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐹 ∈ (Fil‘𝑋))
15 simpr 476 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐵𝐹)
167adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → (𝐴𝐵) ⊆ 𝑋)
17 ssun2 3810 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1817a1i 11 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐵 ⊆ (𝐴𝐵))
19 filss 21704 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐵𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝐵 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
2014, 15, 16, 18, 19syl13anc 1368 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → (𝐴𝐵) ∈ 𝐹)
2120ex 449 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐹 → (𝐴𝐵) ∈ 𝐹))
2213, 21jaod 394 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐹𝐵𝐹) → (𝐴𝐵) ∈ 𝐹))
23 ufilb 21757 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
24233adant3 1101 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
2524adantr 480 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
2623ad2ant1 1102 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
27 difun2 4081 . . . . . . . . . . 11 ((𝐵𝐴) ∖ 𝐴) = (𝐵𝐴)
28 uncom 3790 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
2928difeq1i 3757 . . . . . . . . . . 11 ((𝐵𝐴) ∖ 𝐴) = ((𝐴𝐵) ∖ 𝐴)
3027, 29eqtr3i 2675 . . . . . . . . . 10 (𝐵𝐴) = ((𝐴𝐵) ∖ 𝐴)
3130ineq2i 3844 . . . . . . . . 9 (𝑋 ∩ (𝐵𝐴)) = (𝑋 ∩ ((𝐴𝐵) ∖ 𝐴))
32 indifcom 3905 . . . . . . . . 9 (𝐵 ∩ (𝑋𝐴)) = (𝑋 ∩ (𝐵𝐴))
33 indifcom 3905 . . . . . . . . 9 ((𝐴𝐵) ∩ (𝑋𝐴)) = (𝑋 ∩ ((𝐴𝐵) ∖ 𝐴))
3431, 32, 333eqtr4i 2683 . . . . . . . 8 (𝐵 ∩ (𝑋𝐴)) = ((𝐴𝐵) ∩ (𝑋𝐴))
35 filin 21705 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → ((𝐴𝐵) ∩ (𝑋𝐴)) ∈ 𝐹)
362, 35syl3an1 1399 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → ((𝐴𝐵) ∩ (𝑋𝐴)) ∈ 𝐹)
3734, 36syl5eqel 2734 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋𝐴)) ∈ 𝐹)
38 simp13 1113 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐵𝑋)
39 inss1 3866 . . . . . . . 8 (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵
4039a1i 11 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵)
41 filss 21704 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝐵 ∩ (𝑋𝐴)) ∈ 𝐹𝐵𝑋 ∧ (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵)) → 𝐵𝐹)
4226, 37, 38, 40, 41syl13anc 1368 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐵𝐹)
43423expia 1286 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → ((𝑋𝐴) ∈ 𝐹𝐵𝐹))
4425, 43sylbid 230 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (¬ 𝐴𝐹𝐵𝐹))
4544orrd 392 . . 3 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (𝐴𝐹𝐵𝐹))
4645ex 449 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐵) ∈ 𝐹 → (𝐴𝐹𝐵𝐹)))
4722, 46impbid 202 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐹𝐵𝐹) ↔ (𝐴𝐵) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054  wcel 2030  cdif 3604  cun 3605  cin 3606  wss 3607  cfv 5926  Filcfil 21696  UFilcufil 21750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-fbas 19791  df-fil 21697  df-ufil 21752
This theorem is referenced by: (None)
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