New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > setswith | Unicode version |
Description: Two ways to express the class of all sets that contain . (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
setswith | Sk k |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 4111 | . . . . . . 7 | |
2 | opkeq1 4059 | . . . . . . . 8 | |
3 | 2 | eleq1d 2419 | . . . . . . 7 Sk Sk |
4 | 1, 3 | rexsn 3768 | . . . . . 6 Sk Sk |
5 | vex 2862 | . . . . . . 7 | |
6 | elssetkg 4269 | . . . . . . 7 Sk | |
7 | 5, 6 | mpan2 652 | . . . . . 6 Sk |
8 | 4, 7 | syl5rbb 249 | . . . . 5 Sk |
9 | 8 | abbidv 2467 | . . . 4 Sk |
10 | df-imak 4189 | . . . 4 Sk k Sk | |
11 | 9, 10 | syl6eqr 2403 | . . 3 Sk k |
12 | iftrue 3668 | . . 3 Sk k Sk k | |
13 | 11, 12 | eqtr4d 2388 | . 2 Sk k |
14 | elex 2867 | . . . . . 6 | |
15 | 14 | con3i 127 | . . . . 5 |
16 | 15 | alrimiv 1631 | . . . 4 |
17 | ab0 3569 | . . . 4 | |
18 | 16, 17 | sylibr 203 | . . 3 |
19 | iffalse 3669 | . . 3 Sk k | |
20 | 18, 19 | eqtr4d 2388 | . 2 Sk k |
21 | 13, 20 | pm2.61i 156 | 1 Sk k |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 176 wal 1540 wceq 1642 wcel 1710 cab 2339 wrex 2615 cvv 2859 c0 3550 cif 3662 csn 3737 copk 4057 kcimak 4179 Sk cssetk 4183 |
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-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-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-if 3663 df-sn 3741 df-pr 3742 df-opk 4058 df-imak 4189 df-ssetk 4193 |
This theorem is referenced by: setswithex 4322 |
Copyright terms: Public domain | W3C validator |