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Mirrors > Home > NFE Home > Th. List > dfidk2 | Unicode version |
Description: Definition of k in terms of Sk. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
dfidk2 | k Sk k Sk |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idkssvvk 4281 | . 2 k k | |
2 | inss1 3475 | . . 3 Sk k Sk Sk | |
3 | ssetkssvvk 4278 | . . 3 Sk k | |
4 | 2, 3 | sstri 3281 | . 2 Sk k Sk k |
5 | eqss 3287 | . . 3 | |
6 | vex 2862 | . . . 4 | |
7 | vex 2862 | . . . 4 | |
8 | opkelidkg 4274 | . . . 4 k | |
9 | 6, 7, 8 | mp2an 653 | . . 3 k |
10 | elin 3219 | . . . 4 Sk k Sk Sk k Sk | |
11 | opkelssetkg 4268 | . . . . . 6 Sk | |
12 | 6, 7, 11 | mp2an 653 | . . . . 5 Sk |
13 | 6, 7 | opkelcnvk 4250 | . . . . . 6 k Sk Sk |
14 | opkelssetkg 4268 | . . . . . . 7 Sk | |
15 | 7, 6, 14 | mp2an 653 | . . . . . 6 Sk |
16 | 13, 15 | bitri 240 | . . . . 5 k Sk |
17 | 12, 16 | anbi12i 678 | . . . 4 Sk k Sk |
18 | 10, 17 | bitri 240 | . . 3 Sk k Sk |
19 | 5, 9, 18 | 3bitr4i 268 | . 2 k Sk k Sk |
20 | 1, 4, 19 | eqrelkriiv 4213 | 1 k Sk k Sk |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 wceq 1642 wcel 1710 cvv 2859 cin 3208 wss 3257 copk 4057 k cxpk 4174 kccnvk 4175 Sk cssetk 4183 k cidk 4184 |
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-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-xpk 4185 df-cnvk 4186 df-ssetk 4193 df-idk 4195 |
This theorem is referenced by: idkex 4314 |
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