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Mirrors > Home > NFE Home > Th. List > clos1ex | Unicode version |
Description: The closure of a set under a set is a set. (Contributed by SF, 11-Feb-2015.) |
Ref | Expression |
---|---|
clos1ex.1 |
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clos1ex.2 |
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Ref | Expression |
---|---|
clos1ex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clos1 5873 |
. 2
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2 | elin 3219 |
. . . . . 6
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3 | elimasn 5019 |
. . . . . . . 8
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4 | df-br 4640 |
. . . . . . . 8
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5 | clos1ex.1 |
. . . . . . . . 9
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6 | vex 2862 |
. . . . . . . . 9
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7 | 5, 6 | brsset 4758 |
. . . . . . . 8
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8 | 3, 4, 7 | 3bitr2i 264 |
. . . . . . 7
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9 | elfix 5787 |
. . . . . . . 8
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10 | brco 4883 |
. . . . . . . . 9
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11 | vex 2862 |
. . . . . . . . . . . . 13
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12 | 6, 11 | brimage 5793 |
. . . . . . . . . . . 12
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13 | 12 | anbi1i 676 |
. . . . . . . . . . 11
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14 | 13 | exbii 1582 |
. . . . . . . . . 10
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15 | clos1ex.2 |
. . . . . . . . . . . 12
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16 | 15, 6 | imaex 4747 |
. . . . . . . . . . 11
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17 | breq1 4642 |
. . . . . . . . . . . 12
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18 | 16, 6 | brsset 4758 |
. . . . . . . . . . . 12
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19 | 17, 18 | syl6bb 252 |
. . . . . . . . . . 11
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20 | 16, 19 | ceqsexv 2894 |
. . . . . . . . . 10
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21 | 14, 20 | bitri 240 |
. . . . . . . . 9
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22 | 10, 21 | bitri 240 |
. . . . . . . 8
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23 | 9, 22 | bitri 240 |
. . . . . . 7
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24 | 8, 23 | anbi12i 678 |
. . . . . 6
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25 | 2, 24 | bitri 240 |
. . . . 5
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26 | 25 | abbi2i 2464 |
. . . 4
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27 | ssetex 4744 |
. . . . . 6
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28 | snex 4111 |
. . . . . 6
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29 | 27, 28 | imaex 4747 |
. . . . 5
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30 | 15 | imageex 5801 |
. . . . . . 7
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31 | 27, 30 | coex 4750 |
. . . . . 6
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32 | 31 | fixex 5789 |
. . . . 5
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33 | 29, 32 | inex 4105 |
. . . 4
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34 | 26, 33 | eqeltrri 2424 |
. . 3
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35 | 34 | intex 4320 |
. 2
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36 | 1, 35 | eqeltri 2423 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-2nd 4797 df-txp 5736 df-fix 5740 df-ins2 5750 df-ins3 5752 df-image 5754 df-clos1 5873 |
This theorem is referenced by: clos1exg 5877 clos1induct 5880 clos1basesuc 5882 sbthlem1 6203 spacval 6282 fnspac 6283 nchoicelem11 6299 nchoicelem16 6304 |
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