![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ssfii | Structured version Visualization version GIF version |
Description: Any element of a set 𝐴 is the intersection of a finite subset of 𝐴. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
ssfii | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | intsn 4874 | . . . 4 ⊢ ∩ {𝑥} = 𝑥 |
3 | simpl 486 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ 𝑉) | |
4 | simpr 488 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
5 | 4 | snssd 4702 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴) |
6 | 1 | snnz 4672 | . . . . . 6 ⊢ {𝑥} ≠ ∅ |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ≠ ∅) |
8 | snfi 8577 | . . . . . 6 ⊢ {𝑥} ∈ Fin | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ Fin) |
10 | elfir 8863 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝑥} ⊆ 𝐴 ∧ {𝑥} ≠ ∅ ∧ {𝑥} ∈ Fin)) → ∩ {𝑥} ∈ (fi‘𝐴)) | |
11 | 3, 5, 7, 9, 10 | syl13anc 1369 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ∩ {𝑥} ∈ (fi‘𝐴)) |
12 | 2, 11 | eqeltrrid 2895 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (fi‘𝐴)) |
13 | 12 | ex 416 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (fi‘𝐴))) |
14 | 13 | ssrdv 3921 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 {csn 4525 ∩ cint 4838 ‘cfv 6324 Fincfn 8492 ficfi 8858 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-1o 8085 df-en 8493 df-fin 8496 df-fi 8859 |
This theorem is referenced by: fieq0 8869 dffi2 8871 inficl 8873 fiuni 8876 dffi3 8879 inffien 9474 fictb 9656 ordtbas2 21796 ordtbas 21797 ordtopn1 21799 ordtopn2 21800 leordtval2 21817 subbascn 21859 2ndcsb 22054 ptbasfi 22186 xkoopn 22194 fsubbas 22472 fbunfip 22474 isufil2 22513 ufileu 22524 filufint 22525 fmfnfmlem4 22562 fmfnfm 22563 hausflim 22586 flimclslem 22589 fclsfnflim 22632 flimfnfcls 22633 fclscmp 22635 alexsubb 22651 alexsubALTlem4 22655 ordtconnlem1 31277 topjoin 33826 |
Copyright terms: Public domain | W3C validator |