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Theorem ufprim 22517
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 22512 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
213ad2ant1 1130 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → 𝐹 ∈ (Fil‘𝑋))
32adantr 484 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐹 ∈ (Fil‘𝑋))
4 simpr 488 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐴𝐹)
5 unss 4114 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
65biimpi 219 . . . . . . 7 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
763adant1 1127 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
87adantr 484 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ⊆ 𝑋)
9 ssun1 4102 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
109a1i 11 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝐴𝐵))
11 filss 22461 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝐴 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
123, 4, 8, 10, 11syl13anc 1369 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ∈ 𝐹)
1312ex 416 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐹 → (𝐴𝐵) ∈ 𝐹))
142adantr 484 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐹 ∈ (Fil‘𝑋))
15 simpr 488 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐵𝐹)
167adantr 484 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → (𝐴𝐵) ⊆ 𝑋)
17 ssun2 4103 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1817a1i 11 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → 𝐵 ⊆ (𝐴𝐵))
19 filss 22461 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐵𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝐵 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
2014, 15, 16, 18, 19syl13anc 1369 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐵𝐹) → (𝐴𝐵) ∈ 𝐹)
2120ex 416 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐹 → (𝐴𝐵) ∈ 𝐹))
2213, 21jaod 856 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐹𝐵𝐹) → (𝐴𝐵) ∈ 𝐹))
23 ufilb 22514 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
24233adant3 1129 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
2524adantr 484 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (¬ 𝐴𝐹 ↔ (𝑋𝐴) ∈ 𝐹))
2623ad2ant1 1130 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
27 difun2 4390 . . . . . . . . . . 11 ((𝐵𝐴) ∖ 𝐴) = (𝐵𝐴)
28 uncom 4083 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
2928difeq1i 4049 . . . . . . . . . . 11 ((𝐵𝐴) ∖ 𝐴) = ((𝐴𝐵) ∖ 𝐴)
3027, 29eqtr3i 2826 . . . . . . . . . 10 (𝐵𝐴) = ((𝐴𝐵) ∖ 𝐴)
3130ineq2i 4139 . . . . . . . . 9 (𝑋 ∩ (𝐵𝐴)) = (𝑋 ∩ ((𝐴𝐵) ∖ 𝐴))
32 indifcom 4202 . . . . . . . . 9 (𝐵 ∩ (𝑋𝐴)) = (𝑋 ∩ (𝐵𝐴))
33 indifcom 4202 . . . . . . . . 9 ((𝐴𝐵) ∩ (𝑋𝐴)) = (𝑋 ∩ ((𝐴𝐵) ∖ 𝐴))
3431, 32, 333eqtr4i 2834 . . . . . . . 8 (𝐵 ∩ (𝑋𝐴)) = ((𝐴𝐵) ∩ (𝑋𝐴))
35 filin 22462 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → ((𝐴𝐵) ∩ (𝑋𝐴)) ∈ 𝐹)
362, 35syl3an1 1160 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → ((𝐴𝐵) ∩ (𝑋𝐴)) ∈ 𝐹)
3734, 36eqeltrid 2897 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋𝐴)) ∈ 𝐹)
38 simp13 1202 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐵𝑋)
39 inss1 4158 . . . . . . . 8 (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵
4039a1i 11 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵)
41 filss 22461 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝐵 ∩ (𝑋𝐴)) ∈ 𝐹𝐵𝑋 ∧ (𝐵 ∩ (𝑋𝐴)) ⊆ 𝐵)) → 𝐵𝐹)
4226, 37, 38, 40, 41syl13anc 1369 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹 ∧ (𝑋𝐴) ∈ 𝐹) → 𝐵𝐹)
43423expia 1118 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → ((𝑋𝐴) ∈ 𝐹𝐵𝐹))
4425, 43sylbid 243 . . . 4 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (¬ 𝐴𝐹𝐵𝐹))
4544orrd 860 . . 3 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (𝐴𝐹𝐵𝐹))
4645ex 416 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐵) ∈ 𝐹 → (𝐴𝐹𝐵𝐹)))
4722, 46impbid 215 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐹𝐵𝐹) ↔ (𝐴𝐵) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084  wcel 2112  cdif 3881  cun 3882  cin 3883  wss 3884  cfv 6328  Filcfil 22453  UFilcufil 22507
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-fbas 20091  df-fil 22454  df-ufil 22509
This theorem is referenced by: (None)
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