| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fisn | Structured version Visualization version GIF version | ||
| Description: A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.) |
| Ref | Expression |
|---|---|
| fisn | ⊢ (fi‘{𝐴}) = {𝐴} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elsni 4643 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
| 2 | elsni 4643 | . . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴) | |
| 3 | 1, 2 | ineqan12d 4222 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝐴)) |
| 4 | inidm 4227 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
| 5 | 3, 4 | eqtrdi 2793 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥 ∩ 𝑦) = 𝐴) |
| 6 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 7 | 6 | inex1 5317 | . . . . 5 ⊢ (𝑥 ∩ 𝑦) ∈ V |
| 8 | 7 | elsn 4641 | . . . 4 ⊢ ((𝑥 ∩ 𝑦) ∈ {𝐴} ↔ (𝑥 ∩ 𝑦) = 𝐴) |
| 9 | 5, 8 | sylibr 234 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥 ∩ 𝑦) ∈ {𝐴}) |
| 10 | 9 | rgen2 3199 | . 2 ⊢ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥 ∩ 𝑦) ∈ {𝐴} |
| 11 | snex 5436 | . . 3 ⊢ {𝐴} ∈ V | |
| 12 | inficl 9465 | . . 3 ⊢ ({𝐴} ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥 ∩ 𝑦) ∈ {𝐴} ↔ (fi‘{𝐴}) = {𝐴})) | |
| 13 | 11, 12 | ax-mp 5 | . 2 ⊢ (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥 ∩ 𝑦) ∈ {𝐴} ↔ (fi‘{𝐴}) = {𝐴}) |
| 14 | 10, 13 | mpbi 230 | 1 ⊢ (fi‘{𝐴}) = {𝐴} |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∩ cin 3950 {csn 4626 ‘cfv 6561 ficfi 9450 |
| 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 ax-un 7755 |
| 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-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 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-lim 6389 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-om 7888 df-1o 8506 df-2o 8507 df-en 8986 df-fin 8989 df-fi 9451 |
| This theorem is referenced by: ordtbas 23200 |
| Copyright terms: Public domain | W3C validator |