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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 3499 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | intsn 4914 | . . . 4 ⊢ ∩ {𝑥} = 𝑥 |
3 | simpl 485 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ 𝑉) | |
4 | simpr 487 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
5 | 4 | snssd 4744 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴) |
6 | 1 | snnz 4713 | . . . . . 6 ⊢ {𝑥} ≠ ∅ |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ≠ ∅) |
8 | snfi 8596 | . . . . . 6 ⊢ {𝑥} ∈ Fin | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ Fin) |
10 | elfir 8881 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝑥} ⊆ 𝐴 ∧ {𝑥} ≠ ∅ ∧ {𝑥} ∈ Fin)) → ∩ {𝑥} ∈ (fi‘𝐴)) | |
11 | 3, 5, 7, 9, 10 | syl13anc 1368 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ∩ {𝑥} ∈ (fi‘𝐴)) |
12 | 2, 11 | eqeltrrid 2920 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (fi‘𝐴)) |
13 | 12 | ex 415 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (fi‘𝐴))) |
14 | 13 | ssrdv 3975 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ≠ wne 3018 ⊆ wss 3938 ∅c0 4293 {csn 4569 ∩ cint 4878 ‘cfv 6357 Fincfn 8511 ficfi 8876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1o 8104 df-en 8512 df-fin 8515 df-fi 8877 |
This theorem is referenced by: fieq0 8887 dffi2 8889 inficl 8891 fiuni 8894 dffi3 8897 inffien 9491 fictb 9669 ordtbas2 21801 ordtbas 21802 ordtopn1 21804 ordtopn2 21805 leordtval2 21822 subbascn 21864 2ndcsb 22059 ptbasfi 22191 xkoopn 22199 fsubbas 22477 fbunfip 22479 isufil2 22518 ufileu 22529 filufint 22530 fmfnfmlem4 22567 fmfnfm 22568 hausflim 22591 flimclslem 22594 fclsfnflim 22637 flimfnfcls 22638 fclscmp 22640 alexsubb 22656 alexsubALTlem4 22660 ordtconnlem1 31169 topjoin 33715 |
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