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Mirrors > Home > NFE Home > Th. List > elp6 | Unicode version |
Description: Membership in the P6 operator. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
elp6 | P6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3744 | . . . . . 6 | |
2 | 1 | sneqd 3746 | . . . . 5 |
3 | 2 | xpkeq2d 4205 | . . . 4 k k |
4 | 3 | sseq1d 3298 | . . 3 k k |
5 | df-p6 4191 | . . 3 P6 k | |
6 | 4, 5 | elab2g 2987 | . 2 P6 k |
7 | xpkssvvk 4210 | . . . 4 k k | |
8 | ssrelk 4211 | . . . 4 k k k k | |
9 | 7, 8 | ax-mp 5 | . . 3 k k |
10 | vex 2862 | . . . . . . . . 9 | |
11 | vex 2862 | . . . . . . . . 9 | |
12 | 10, 11 | opkelxpk 4248 | . . . . . . . 8 k |
13 | 10 | biantrur 492 | . . . . . . . 8 |
14 | df-sn 3741 | . . . . . . . . 9 | |
15 | 14 | abeq2i 2460 | . . . . . . . 8 |
16 | 12, 13, 15 | 3bitr2i 264 | . . . . . . 7 k |
17 | 16 | imbi1i 315 | . . . . . 6 k |
18 | 17 | albii 1566 | . . . . 5 k |
19 | snex 4111 | . . . . . 6 | |
20 | opkeq2 4060 | . . . . . . 7 | |
21 | 20 | eleq1d 2419 | . . . . . 6 |
22 | 19, 21 | ceqsalv 2885 | . . . . 5 |
23 | 18, 22 | bitri 240 | . . . 4 k |
24 | 23 | albii 1566 | . . 3 k |
25 | 9, 24 | bitri 240 | . 2 k |
26 | 6, 25 | syl6bb 252 | 1 P6 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wal 1540 wceq 1642 wcel 1710 cvv 2859 wss 3257 csn 3737 copk 4057 k cxpk 4174 P6 cp6 4178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-rex 2620 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-p6 4191 |
This theorem is referenced by: p6exg 4290 dfuni12 4291 dfimak2 4298 |
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