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Mirrors > Home > MPE Home > Th. List > isufil | Structured version Visualization version GIF version |
Description: The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.) |
Ref | Expression |
---|---|
isufil | ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ufil 23896 | . 2 ⊢ UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧)}) | |
2 | pweq 4621 | . . . 4 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋) | |
3 | 2 | adantr 479 | . . 3 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋) |
4 | eleq2 2815 | . . . . 5 ⊢ (𝑧 = 𝐹 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹)) | |
5 | 4 | adantl 480 | . . . 4 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹)) |
6 | difeq1 4114 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ∖ 𝑥) = (𝑋 ∖ 𝑥)) | |
7 | eleq12 2816 | . . . . 5 ⊢ (((𝑦 ∖ 𝑥) = (𝑋 ∖ 𝑥) ∧ 𝑧 = 𝐹) → ((𝑦 ∖ 𝑥) ∈ 𝑧 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹)) | |
8 | 6, 7 | sylan 578 | . . . 4 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → ((𝑦 ∖ 𝑥) ∈ 𝑧 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹)) |
9 | 5, 8 | orbi12d 916 | . . 3 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → ((𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧) ↔ (𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) |
10 | 3, 9 | raleqbidv 3330 | . 2 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) |
11 | fveq2 6901 | . 2 ⊢ (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋)) | |
12 | fvex 6914 | . 2 ⊢ (Fil‘𝑦) ∈ V | |
13 | elfvdm 6938 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil) | |
14 | 1, 10, 11, 12, 13 | elmptrab2 23823 | 1 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∖ cdif 3944 𝒫 cpw 4607 dom cdm 5682 ‘cfv 6554 Filcfil 23840 UFilcufil 23894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fv 6562 df-ufil 23896 |
This theorem is referenced by: ufilfil 23899 ufilss 23900 isufil2 23903 trufil 23905 fixufil 23917 fin1aufil 23927 |
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