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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2741 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | simpr 109 | . . . . 5 | |
4 | eldifsni 3712 | . . . . . . . 8 | |
5 | 4 | adantr 274 | . . . . . . 7 |
6 | eldifi 3249 | . . . . . . . . . 10 | |
7 | 6 | elin2d 3317 | . . . . . . . . 9 |
8 | 7 | adantr 274 | . . . . . . . 8 |
9 | fin0 6863 | . . . . . . . 8 | |
10 | 8, 9 | syl 14 | . . . . . . 7 |
11 | 5, 10 | mpbid 146 | . . . . . 6 |
12 | inteximm 4135 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | 3, 13 | eqeltrd 2247 | . . . 4 |
15 | 14 | rexlimiva 2582 | . . 3 |
16 | 15 | a1i 9 | . 2 |
17 | elfi 6948 | . . . 4 | |
18 | vprc 4121 | . . . . . . . . . . 11 | |
19 | elsni 3601 | . . . . . . . . . . . . . 14 | |
20 | 19 | inteqd 3836 | . . . . . . . . . . . . 13 |
21 | int0 3845 | . . . . . . . . . . . . 13 | |
22 | 20, 21 | eqtrdi 2219 | . . . . . . . . . . . 12 |
23 | 22 | eleq1d 2239 | . . . . . . . . . . 11 |
24 | 18, 23 | mtbiri 670 | . . . . . . . . . 10 |
25 | simpr 109 | . . . . . . . . . . 11 | |
26 | simpll 524 | . . . . . . . . . . 11 | |
27 | 25, 26 | eqeltrrd 2248 | . . . . . . . . . 10 |
28 | 24, 27 | nsyl3 621 | . . . . . . . . 9 |
29 | 28 | biantrud 302 | . . . . . . . 8 |
30 | eldif 3130 | . . . . . . . 8 | |
31 | 29, 30 | bitr4di 197 | . . . . . . 7 |
32 | 31 | pm5.32da 449 | . . . . . 6 |
33 | ancom 264 | . . . . . 6 | |
34 | ancom 264 | . . . . . 6 | |
35 | 32, 33, 34 | 3bitr4g 222 | . . . . 5 |
36 | 35 | rexbidv2 2473 | . . . 4 |
37 | 17, 36 | bitrd 187 | . . 3 |
38 | 37 | expcom 115 | . 2 |
39 | 2, 16, 38 | pm5.21ndd 700 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 wne 2340 wrex 2449 cvv 2730 cdif 3118 cin 3120 c0 3414 cpw 3566 csn 3583 cint 3831 cfv 5198 cfn 6718 cfi 6945 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-er 6513 df-en 6719 df-fin 6721 df-fi 6946 |
This theorem is referenced by: fiuni 6955 fifo 6957 |
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