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Mirrors > Home > ILE Home > Th. List > elfi2 | Unicode version |
Description: The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
elfi2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2700 |
. . 3
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2 | 1 | a1i 9 |
. 2
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3 | simpr 109 |
. . . . 5
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4 | eldifsni 3660 |
. . . . . . . 8
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5 | 4 | adantr 274 |
. . . . . . 7
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6 | eldifi 3203 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | elin2d 3271 |
. . . . . . . . 9
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8 | 7 | adantr 274 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | fin0 6787 |
. . . . . . . 8
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10 | 8, 9 | syl 14 |
. . . . . . 7
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11 | 5, 10 | mpbid 146 |
. . . . . 6
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12 | inteximm 4082 |
. . . . . 6
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13 | 11, 12 | syl 14 |
. . . . 5
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14 | 3, 13 | eqeltrd 2217 |
. . . 4
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15 | 14 | rexlimiva 2547 |
. . 3
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16 | 15 | a1i 9 |
. 2
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17 | elfi 6867 |
. . . 4
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18 | vprc 4068 |
. . . . . . . . . . 11
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19 | elsni 3550 |
. . . . . . . . . . . . . 14
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20 | 19 | inteqd 3784 |
. . . . . . . . . . . . 13
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21 | int0 3793 |
. . . . . . . . . . . . 13
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22 | 20, 21 | eqtrdi 2189 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | eleq1d 2209 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 18, 23 | mtbiri 665 |
. . . . . . . . . 10
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25 | simpr 109 |
. . . . . . . . . . 11
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26 | simpll 519 |
. . . . . . . . . . 11
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27 | 25, 26 | eqeltrrd 2218 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 24, 27 | nsyl3 616 |
. . . . . . . . 9
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29 | 28 | biantrud 302 |
. . . . . . . 8
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30 | eldif 3085 |
. . . . . . . 8
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31 | 29, 30 | syl6bbr 197 |
. . . . . . 7
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32 | 31 | pm5.32da 448 |
. . . . . 6
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33 | ancom 264 |
. . . . . 6
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34 | ancom 264 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
35 | 32, 33, 34 | 3bitr4g 222 |
. . . . 5
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36 | 35 | rexbidv2 2441 |
. . . 4
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37 | 17, 36 | bitrd 187 |
. . 3
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38 | 37 | expcom 115 |
. 2
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39 | 2, 16, 38 | pm5.21ndd 695 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-er 6437 df-en 6643 df-fin 6645 df-fi 6865 |
This theorem is referenced by: fiuni 6874 fifo 6876 |
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