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Mirrors > Home > MPE Home > Th. List > inficl | Structured version Visualization version Unicode version |
Description: A set which is closed under pairwise intersection is closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
inficl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssfii 7933 |
. . 3
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2 | eqimss2 3485 |
. . . . . . . 8
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3 | 2 | biantrurd 511 |
. . . . . . 7
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4 | eleq2 2518 |
. . . . . . . . 9
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5 | 4 | raleqbi1dv 2995 |
. . . . . . . 8
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6 | 5 | raleqbi1dv 2995 |
. . . . . . 7
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7 | 3, 6 | bitr3d 259 |
. . . . . 6
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8 | 7 | elabg 3186 |
. . . . 5
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9 | intss1 4249 |
. . . . 5
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10 | 8, 9 | syl6bir 233 |
. . . 4
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11 | dffi2 7937 |
. . . . 5
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12 | 11 | sseq1d 3459 |
. . . 4
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13 | 10, 12 | sylibrd 238 |
. . 3
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14 | eqss 3447 |
. . . 4
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15 | 14 | simplbi2com 633 |
. . 3
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16 | 1, 13, 15 | sylsyld 58 |
. 2
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17 | fiin 7936 |
. . . 4
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18 | 17 | rgen2a 2815 |
. . 3
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19 | eleq2 2518 |
. . . . 5
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20 | 19 | raleqbi1dv 2995 |
. . . 4
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21 | 20 | raleqbi1dv 2995 |
. . 3
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22 | 18, 21 | mpbii 215 |
. 2
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23 | 16, 22 | impbid1 207 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-oadd 7186 df-er 7363 df-en 7570 df-fin 7573 df-fi 7925 |
This theorem is referenced by: fipwuni 7940 fisn 7941 fitop 19930 ordtbaslem 20204 ptbasin2 20593 filfi 20874 fmfnfmlem3 20971 ustuqtop2 21257 ldgenpisys 28988 |
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