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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 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | intsn 4448 | . . . 4 ⊢ ∩ {𝑥} = 𝑥 |
3 | simpl 472 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ 𝑉) | |
4 | simpr 476 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
5 | 4 | snssd 4281 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴) |
6 | 1 | snnz 4252 | . . . . . 6 ⊢ {𝑥} ≠ ∅ |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ≠ ∅) |
8 | snfi 7923 | . . . . . 6 ⊢ {𝑥} ∈ Fin | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ Fin) |
10 | elfir 8204 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝑥} ⊆ 𝐴 ∧ {𝑥} ≠ ∅ ∧ {𝑥} ∈ Fin)) → ∩ {𝑥} ∈ (fi‘𝐴)) | |
11 | 3, 5, 7, 9, 10 | syl13anc 1320 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ∩ {𝑥} ∈ (fi‘𝐴)) |
12 | 2, 11 | syl5eqelr 2693 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (fi‘𝐴)) |
13 | 12 | ex 449 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (fi‘𝐴))) |
14 | 13 | ssrdv 3574 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 {csn 4125 ∩ cint 4410 ‘cfv 5804 Fincfn 7841 ficfi 8199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-en 7842 df-fin 7845 df-fi 8200 |
This theorem is referenced by: fieq0 8210 dffi2 8212 inficl 8214 fiuni 8217 dffi3 8220 inffien 8769 fictb 8950 ordtbas2 20805 ordtbas 20806 ordtopn1 20808 ordtopn2 20809 leordtval2 20826 subbascn 20868 2ndcsb 21062 ptbasfi 21194 xkoopn 21202 fsubbas 21481 fbunfip 21483 isufil2 21522 ufileu 21533 filufint 21534 fmfnfmlem4 21571 fmfnfm 21572 hausflim 21595 flimclslem 21598 fclsfnflim 21641 flimfnfcls 21642 fclscmp 21644 alexsubb 21660 alexsubALTlem4 21664 ordtconlem1 29298 topjoin 31530 |
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