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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 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | intsn 4989 | . . . 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 4814 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴) |
6 | 1 | snnz 4781 | . . . . . 6 ⊢ {𝑥} ≠ ∅ |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ≠ ∅) |
8 | snfi 9082 | . . . . . 6 ⊢ {𝑥} ∈ Fin | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ Fin) |
10 | elfir 9453 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝑥} ⊆ 𝐴 ∧ {𝑥} ≠ ∅ ∧ {𝑥} ∈ Fin)) → ∩ {𝑥} ∈ (fi‘𝐴)) | |
11 | 3, 5, 7, 9, 10 | syl13anc 1371 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ∩ {𝑥} ∈ (fi‘𝐴)) |
12 | 2, 11 | eqeltrrid 2844 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (fi‘𝐴)) |
13 | 12 | ex 412 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (fi‘𝐴))) |
14 | 13 | ssrdv 4001 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 {csn 4631 ∩ cint 4951 ‘cfv 6563 Fincfn 8984 ficfi 9448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-en 8985 df-fin 8988 df-fi 9449 |
This theorem is referenced by: fieq0 9459 dffi2 9461 inficl 9463 fiuni 9466 dffi3 9469 inffien 10101 fictb 10282 ordtbas2 23215 ordtbas 23216 ordtopn1 23218 ordtopn2 23219 leordtval2 23236 subbascn 23278 2ndcsb 23473 ptbasfi 23605 xkoopn 23613 fsubbas 23891 fbunfip 23893 isufil2 23932 ufileu 23943 filufint 23944 fmfnfmlem4 23981 fmfnfm 23982 hausflim 24005 flimclslem 24008 fclsfnflim 24051 flimfnfcls 24052 fclscmp 24054 alexsubb 24070 alexsubALTlem4 24074 ordtconnlem1 33885 topjoin 36348 |
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