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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 3478 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | intsn 4989 | . . . 4 ⊢ ∩ {𝑥} = 𝑥 |
3 | simpl 483 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ 𝑉) | |
4 | simpr 485 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
5 | 4 | snssd 4811 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴) |
6 | 1 | snnz 4779 | . . . . . 6 ⊢ {𝑥} ≠ ∅ |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ≠ ∅) |
8 | snfi 9040 | . . . . . 6 ⊢ {𝑥} ∈ Fin | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ Fin) |
10 | elfir 9406 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝑥} ⊆ 𝐴 ∧ {𝑥} ≠ ∅ ∧ {𝑥} ∈ Fin)) → ∩ {𝑥} ∈ (fi‘𝐴)) | |
11 | 3, 5, 7, 9, 10 | syl13anc 1372 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ∩ {𝑥} ∈ (fi‘𝐴)) |
12 | 2, 11 | eqeltrrid 2838 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (fi‘𝐴)) |
13 | 12 | ex 413 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (fi‘𝐴))) |
14 | 13 | ssrdv 3987 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3947 ∅c0 4321 {csn 4627 ∩ cint 4949 ‘cfv 6540 Fincfn 8935 ficfi 9401 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7852 df-1o 8462 df-en 8936 df-fin 8939 df-fi 9402 |
This theorem is referenced by: fieq0 9412 dffi2 9414 inficl 9416 fiuni 9419 dffi3 9422 inffien 10054 fictb 10236 ordtbas2 22686 ordtbas 22687 ordtopn1 22689 ordtopn2 22690 leordtval2 22707 subbascn 22749 2ndcsb 22944 ptbasfi 23076 xkoopn 23084 fsubbas 23362 fbunfip 23364 isufil2 23403 ufileu 23414 filufint 23415 fmfnfmlem4 23452 fmfnfm 23453 hausflim 23476 flimclslem 23479 fclsfnflim 23522 flimfnfcls 23523 fclscmp 23525 alexsubb 23541 alexsubALTlem4 23545 ordtconnlem1 32892 topjoin 35238 |
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