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| Mirrors > Home > MPE Home > Th. List > ufildom1 | Structured version Visualization version GIF version | ||
| Description: An ultrafilter is generated by at most one element (because free ultrafilters have no generators and fixed ultrafilters have exactly one). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
| Ref | Expression |
|---|---|
| ufildom1 | ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ≼ 1o) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | breq1 5146 | . 2 ⊢ (∩ 𝐹 = ∅ → (∩ 𝐹 ≼ 1o ↔ ∅ ≼ 1o)) | |
| 2 | uffixsn 23933 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ∈ 𝐹) | |
| 3 | intss1 4963 | . . . . . . . . 9 ⊢ ({𝑥} ∈ 𝐹 → ∩ 𝐹 ⊆ {𝑥}) | |
| 4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 ⊆ {𝑥}) |
| 5 | simpr 484 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → 𝑥 ∈ ∩ 𝐹) | |
| 6 | 5 | snssd 4809 | . . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → {𝑥} ⊆ ∩ 𝐹) |
| 7 | 4, 6 | eqssd 4001 | . . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ ∩ 𝐹) → ∩ 𝐹 = {𝑥}) |
| 8 | 7 | ex 412 | . . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ ∩ 𝐹 → ∩ 𝐹 = {𝑥})) |
| 9 | 8 | eximdv 1917 | . . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∃𝑥 𝑥 ∈ ∩ 𝐹 → ∃𝑥∩ 𝐹 = {𝑥})) |
| 10 | n0 4353 | . . . . 5 ⊢ (∩ 𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ∩ 𝐹) | |
| 11 | en1 9064 | . . . . 5 ⊢ (∩ 𝐹 ≈ 1o ↔ ∃𝑥∩ 𝐹 = {𝑥}) | |
| 12 | 9, 10, 11 | 3imtr4g 296 | . . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 ≠ ∅ → ∩ 𝐹 ≈ 1o)) |
| 13 | 12 | imp 406 | . . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≈ 1o) |
| 14 | endom 9019 | . . 3 ⊢ (∩ 𝐹 ≈ 1o → ∩ 𝐹 ≼ 1o) | |
| 15 | 13, 14 | syl 17 | . 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 ≠ ∅) → ∩ 𝐹 ≼ 1o) |
| 16 | 1on 8518 | . . 3 ⊢ 1o ∈ On | |
| 17 | 0domg 9140 | . . 3 ⊢ (1o ∈ On → ∅ ≼ 1o) | |
| 18 | 16, 17 | mp1i 13 | . 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∅ ≼ 1o) |
| 19 | 1, 15, 18 | pm2.61ne 3027 | 1 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ≼ 1o) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 {csn 4626 ∩ cint 4946 class class class wbr 5143 Oncon0 6384 ‘cfv 6561 1oc1o 8499 ≈ cen 8982 ≼ cdom 8983 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-3or 1088 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-reu 3381 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-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1o 8506 df-en 8986 df-dom 8987 df-fbas 21361 df-fg 21362 df-fil 23854 df-ufil 23909 |
| This theorem is referenced by: (None) |
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