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Mirrors > Home > MPE Home > Th. List > intex | Structured version Visualization version Unicode version |
Description: The intersection of a nonempty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.) |
Ref | Expression |
---|---|
intex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3931 | . . 3 | |
2 | intss1 4492 | . . . . 5 | |
3 | vex 3203 | . . . . . 6 | |
4 | 3 | ssex 4802 | . . . . 5 |
5 | 2, 4 | syl 17 | . . . 4 |
6 | 5 | exlimiv 1858 | . . 3 |
7 | 1, 6 | sylbi 207 | . 2 |
8 | vprc 4796 | . . . 4 | |
9 | inteq 4478 | . . . . . 6 | |
10 | int0 4490 | . . . . . 6 | |
11 | 9, 10 | syl6eq 2672 | . . . . 5 |
12 | 11 | eleq1d 2686 | . . . 4 |
13 | 8, 12 | mtbiri 317 | . . 3 |
14 | 13 | necon2ai 2823 | . 2 |
15 | 7, 14 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wceq 1483 wex 1704 wcel 1990 wne 2794 cvv 3200 wss 3574 c0 3915 cint 4475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-int 4476 |
This theorem is referenced by: intnex 4821 intexab 4822 iinexg 4824 onint0 6996 onintrab 7001 onmindif2 7012 fival 8318 elfi2 8320 elfir 8321 dffi2 8329 elfiun 8336 fifo 8338 tz9.1c 8606 tz9.12lem1 8650 tz9.12lem3 8652 rankf 8657 cardf2 8769 cardval3 8778 cardid2 8779 cardcf 9074 cflim2 9085 intwun 9557 wuncval 9564 inttsk 9596 intgru 9636 gruina 9640 dfrtrcl2 13802 mremre 16264 mrcval 16270 asplss 19329 aspsubrg 19331 toponmre 20897 subbascn 21058 insiga 30200 sigagenval 30203 sigagensiga 30204 dmsigagen 30207 dfon2lem8 31695 dfon2lem9 31696 bj-snmoore 33068 igenval 33860 pclvalN 35176 elrfi 37257 ismrcd1 37261 mzpval 37295 dmmzp 37296 salgenval 40541 intsal 40548 |
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