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Mirrors > Home > NFE Home > Th. List > ins3kss | Unicode version |
Description: Subset law for Ins3k ![]() |
Ref | Expression |
---|---|
ins3kss |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 |
. . . . 5
![]() ![]() ![]() ![]() | |
2 | vex 2862 |
. . . . 5
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3 | opkelins3kg 4252 |
. . . . 5
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4 | 1, 2, 3 | mp2an 653 |
. . . 4
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5 | opkeq12 4061 |
. . . . . . . 8
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6 | vex 2862 |
. . . . . . . . . . 11
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7 | 6 | snel1c 4140 |
. . . . . . . . . 10
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8 | snelpw1 4146 |
. . . . . . . . . 10
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9 | 7, 8 | mpbir 200 |
. . . . . . . . 9
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10 | vex 2862 |
. . . . . . . . . 10
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11 | vex 2862 |
. . . . . . . . . 10
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12 | 10, 11 | opkelxpk 4248 |
. . . . . . . . . 10
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13 | 10, 11, 12 | mpbir2an 886 |
. . . . . . . . 9
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14 | snex 4111 |
. . . . . . . . . 10
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15 | opkex 4113 |
. . . . . . . . . 10
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16 | 14, 15 | opkelxpk 4248 |
. . . . . . . . 9
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17 | 9, 13, 16 | mpbir2an 886 |
. . . . . . . 8
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18 | 5, 17 | syl6eqel 2441 |
. . . . . . 7
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19 | 18 | 3adant3 975 |
. . . . . 6
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20 | 19 | exlimiv 1634 |
. . . . 5
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21 | 20 | exlimivv 1635 |
. . . 4
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22 | 4, 21 | sylbi 187 |
. . 3
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23 | 22 | gen2 1547 |
. 2
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24 | df-ins3k 4188 |
. . . 4
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25 | 24 | opkabssvvki 4209 |
. . 3
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26 | ssrelk 4211 |
. . 3
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27 | 25, 26 | ax-mp 5 |
. 2
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28 | 23, 27 | mpbir 200 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-ins3k 4188 |
This theorem is referenced by: ins3kexg 4306 |
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