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Mirrors > Home > NFE Home > Th. List > ins2kss | Unicode version |
Description: Subset law for Ins2k . (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
ins2kss | Ins2k 1 1c k k |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 | . . . . 5 | |
2 | vex 2862 | . . . . 5 | |
3 | opkelins2kg 4251 | . . . . 5 Ins2k | |
4 | 1, 2, 3 | mp2an 653 | . . . 4 Ins2k |
5 | opkeq12 4061 | . . . . . . . 8 | |
6 | vex 2862 | . . . . . . . . . . 11 | |
7 | 6 | snel1c 4140 | . . . . . . . . . 10 1c |
8 | snelpw1 4146 | . . . . . . . . . 10 1 1c 1c | |
9 | 7, 8 | mpbir 200 | . . . . . . . . 9 1 1c |
10 | vex 2862 | . . . . . . . . . 10 | |
11 | vex 2862 | . . . . . . . . . 10 | |
12 | 10, 11 | opkelxpk 4248 | . . . . . . . . . 10 k |
13 | 10, 11, 12 | mpbir2an 886 | . . . . . . . . 9 k |
14 | snex 4111 | . . . . . . . . . 10 | |
15 | opkex 4113 | . . . . . . . . . 10 | |
16 | 14, 15 | opkelxpk 4248 | . . . . . . . . 9 1 1c k k 1 1c k |
17 | 9, 13, 16 | mpbir2an 886 | . . . . . . . 8 1 1c k k |
18 | 5, 17 | syl6eqel 2441 | . . . . . . 7 1 1c k k |
19 | 18 | 3adant3 975 | . . . . . 6 1 1c k k |
20 | 19 | exlimiv 1634 | . . . . 5 1 1c k k |
21 | 20 | exlimivv 1635 | . . . 4 1 1c k k |
22 | 4, 21 | sylbi 187 | . . 3 Ins2k 1 1c k k |
23 | 22 | gen2 1547 | . 2 Ins2k 1 1c k k |
24 | df-ins2k 4187 | . . . 4 Ins2k | |
25 | 24 | opkabssvvki 4209 | . . 3 Ins2k k |
26 | ssrelk 4211 | . . 3 Ins2k k Ins2k 1 1c k k Ins2k 1 1c k k | |
27 | 25, 26 | ax-mp 5 | . 2 Ins2k 1 1c k k Ins2k 1 1c k k |
28 | 23, 27 | mpbir 200 | 1 Ins2k 1 1c k k |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 w3a 934 wal 1540 wex 1541 wceq 1642 wcel 1710 cvv 2859 wss 3257 csn 3737 copk 4057 1cc1c 4134 1 cpw1 4135 k cxpk 4174 Ins2k cins2k 4176 |
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-pw 3724 df-sn 3741 df-pr 3742 df-opk 4058 df-1c 4136 df-pw1 4137 df-xpk 4185 df-ins2k 4187 |
This theorem is referenced by: ins2kexg 4305 |
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