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| Mirrors > Home > NFE Home > Th. List > el1st | Unicode version | ||
Description: Membership in  . (Contributed by SF,
5-Jan-2015.) |
| Ref | Expression |
|---|---|
| el1st |
       
     |
,,
is distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-1st 4723 |
. . . 4
   | |
| 2 | 1 | eleq2i 2417 |
. . 3
|
| 3 | elopab 4696 |
. . 3
![](wedge.gif "/\"){kind=small}
| |
| 4 | 2, 3 | bitri 240 |
. 2
|
| 5 | excom 1741 |
. . 3
| |
| 6 | excom 1741 |
. . . . 5
| |
| 7 | exancom 1586 |
. . . . . . 7
| |
| 8 | vex 2862 |
. . . . . . . . 9
 | |
| 9 | vex 2862 |
. . . . . . . . 9
| |
| 10 | 8, 9 | opex 4588 |
. . . . . . . 8
|
| 11 | opeq1 4578 |
. . . . . . . . 9
| |
| 12 | 11 | eqeq2d 2364 |
. . . . . . . 8
|
| 13 | 10, 12 | ceqsexv 2894 |
. . . . . . 7
|
| 14 | 7, 13 | bitri 240 |
. . . . . 6
|
| 15 | 14 | exbii 1582 |
. . . . 5
|
| 16 | exdistr 1906 |
. . . . 5
| |
| 17 | 6, 15, 16 | 3bitr3ri 267 |
. . . 4
|
| 18 | 17 | exbii 1582 |
. . 3
|
| 19 | 5, 18 | bitri 240 |
. 2
|
| 20 | 4, 19 | bitri 240 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wb 176
wa 358
wex 1541
wceq 1642
wcel 1710
cop 4561
copab 4622 c1st 4717 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
| This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-opab 4623 df-1st 4723 |
| This theorem is referenced by: br1stg 4730 |
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