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| 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 3483 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | 1 | intsn 4983 | . . . 4 ⊢ ∩ {𝑥} = 𝑥 | 
| 3 | simpl 482 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ 𝑉) | |
| 4 | simpr 484 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
| 5 | 4 | snssd 4808 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴) | 
| 6 | 1 | snnz 4775 | . . . . . 6 ⊢ {𝑥} ≠ ∅ | 
| 7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ≠ ∅) | 
| 8 | snfi 9084 | . . . . . 6 ⊢ {𝑥} ∈ Fin | |
| 9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ Fin) | 
| 10 | elfir 9456 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝑥} ⊆ 𝐴 ∧ {𝑥} ≠ ∅ ∧ {𝑥} ∈ Fin)) → ∩ {𝑥} ∈ (fi‘𝐴)) | |
| 11 | 3, 5, 7, 9, 10 | syl13anc 1373 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ∩ {𝑥} ∈ (fi‘𝐴)) | 
| 12 | 2, 11 | eqeltrrid 2845 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (fi‘𝐴)) | 
| 13 | 12 | ex 412 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (fi‘𝐴))) | 
| 14 | 13 | ssrdv 3988 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ≠ wne 2939 ⊆ wss 3950 ∅c0 4332 {csn 4625 ∩ cint 4945 ‘cfv 6560 Fincfn 8986 ficfi 9451 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-om 7889 df-1o 8507 df-en 8987 df-fin 8990 df-fi 9452 | 
| This theorem is referenced by: fieq0 9462 dffi2 9464 inficl 9466 fiuni 9469 dffi3 9472 inffien 10104 fictb 10285 ordtbas2 23200 ordtbas 23201 ordtopn1 23203 ordtopn2 23204 leordtval2 23221 subbascn 23263 2ndcsb 23458 ptbasfi 23590 xkoopn 23598 fsubbas 23876 fbunfip 23878 isufil2 23917 ufileu 23928 filufint 23929 fmfnfmlem4 23966 fmfnfm 23967 hausflim 23990 flimclslem 23993 fclsfnflim 24036 flimfnfcls 24037 fclscmp 24039 alexsubb 24055 alexsubALTlem4 24059 ordtconnlem1 33924 topjoin 36367 | 
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