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Theorem ufprim 21623
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 21618 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
213ad2ant1 1080 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → 𝐹 ∈ (Fil‘𝑋))
32adantr 481 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐹 ∈ (Fil‘𝑋))
4 simpr 477 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐴𝐹)
5 unss 3765 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
65biimpi 206 . . . . . . 7 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
763adant1 1077 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
87adantr 481 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ⊆ 𝑋)
9 ssun1 3754 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
109a1i 11 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝐴𝐵))
11 filss 21567 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝐴 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
123, 4, 8, 10, 11syl13anc 1325 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ∈ 𝐹)
1312ex 450 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐹 → (𝐴𝐵) ∈ 𝐹))
142adantr 481 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐹 ∈ (Fil‘𝑋))
15 simpr 477 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐵𝐹)
167adantr 481 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → (𝐴𝐵) ⊆ 𝑋)
17 ssun2 3755 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1817a1i 11 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐵 ⊆ (𝐴𝐵))
19 filss 21567 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐵𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝐵 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
2014, 15, 16, 18, 19syl13anc 1325 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → (𝐴𝐵) ∈ 𝐹)
2120ex 450 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐹 → (𝐴𝐵) ∈ 𝐹))
2213, 21jaod 395 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐹𝐵𝐹) → (𝐴𝐵) ∈ 𝐹))
23 ufilb 21620 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
24233adant3 1079 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
2524adantr 481 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
2623ad2ant1 1080 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
27 difun2 4020 . . . . . . . . . . 11 ((𝐵𝐴) ∖ 𝐴) = (𝐵𝐴)
28 uncom 3735 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
2928difeq1i 3702 . . . . . . . . . . 11 ((𝐵𝐴) ∖ 𝐴) = ((𝐴𝐵) ∖ 𝐴)
3027, 29eqtr3i 2645 . . . . . . . . . 10 (𝐵𝐴) = ((𝐴𝐵) ∖ 𝐴)
3130ineq2i 3789 . . . . . . . . 9 (𝑋 ∩ (𝐵𝐴)) = (𝑋 ∩ ((𝐴𝐵) ∖ 𝐴))
32 indifcom 3848 . . . . . . . . 9 (𝐵 ∩ (𝑋𝐴)) = (𝑋 ∩ (𝐵𝐴))
33 indifcom 3848 . . . . . . . . 9 ((𝐴𝐵) ∩ (𝑋𝐴)) = (𝑋 ∩ ((𝐴𝐵) ∖ 𝐴))
3431, 32, 333eqtr4i 2653 . . . . . . . 8 (𝐵 ∩ (𝑋𝐴)) = ((𝐴𝐵) ∩ (𝑋𝐴))
35 filin 21568 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → ((𝐴𝐵) ∩ (𝑋𝐴)) ∈ 𝐹)
362, 35syl3an1 1356 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → ((𝐴𝐵) ∩ (𝑋𝐴)) ∈ 𝐹)
3734, 36syl5eqel 2702 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋𝐴)) ∈ 𝐹)
38 simp13 1091 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐵𝑋)
39 inss1 3811 . . . . . . . 8 (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵
4039a1i 11 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵)
41 filss 21567 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝐵 ∩ (𝑋𝐴)) ∈ 𝐹𝐵𝑋 ∧ (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵)) → 𝐵𝐹)
4226, 37, 38, 40, 41syl13anc 1325 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐵𝐹)
43423expia 1264 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → ((𝑋𝐴) ∈ 𝐹𝐵𝐹))
4425, 43sylbid 230 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (¬ 𝐴𝐹𝐵𝐹))
4544orrd 393 . . 3 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (𝐴𝐹𝐵𝐹))
4645ex 450 . 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 383  wa 384  w3a 1036  wcel 1987  cdif 3552  cun 3553  cin 3554  wss 3555  cfv 5847  Filcfil 21559  UFilcufil 21613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-fbas 19662  df-fil 21560  df-ufil 21615
This theorem is referenced by: (None)
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