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Mirrors > Home > NFE Home > Th. List > clos1induct | Unicode version |
Description: Inductive law for
closure. If the base set is a subset of ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
clos1induct.1 |
![]() ![]() ![]() ![]() |
clos1induct.2 |
![]() ![]() ![]() ![]() |
clos1induct.3 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
clos1induct |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clos1induct.3 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | clos1induct.1 |
. . . . 5
![]() ![]() ![]() ![]() | |
3 | clos1induct.2 |
. . . . 5
![]() ![]() ![]() ![]() | |
4 | 2, 3 | clos1ex 5877 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 1, 4 | eqeltri 2423 |
. . 3
![]() ![]() ![]() ![]() |
6 | inexg 4101 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | mpan2 652 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 1 | clos1base 5879 |
. . 3
![]() ![]() ![]() ![]() |
9 | ssin 3478 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 9 | biimpi 186 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 8, 10 | mpan2 652 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | elima2 4756 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | elin 3220 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 12, 13 | imbi12i 316 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | df-ral 2620 |
. . . . . . . 8
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16 | impexp 433 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 1 | clos1conn 5880 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17 | biantrud 493 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 18 | adantrl 696 |
. . . . . . . . . . 11
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20 | 19 | pm5.74i 236 |
. . . . . . . . . 10
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21 | 16, 20 | bitr3i 242 |
. . . . . . . . 9
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22 | 21 | albii 1566 |
. . . . . . . 8
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23 | 15, 22 | bitri 240 |
. . . . . . 7
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24 | elin 3220 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | ancom 437 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 24, 25 | bitri 240 |
. . . . . . . . . . 11
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27 | 26 | anbi1i 676 |
. . . . . . . . . 10
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28 | anass 630 |
. . . . . . . . . 10
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29 | 27, 28 | bitri 240 |
. . . . . . . . 9
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30 | 29 | imbi1i 315 |
. . . . . . . 8
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31 | 30 | albii 1566 |
. . . . . . 7
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32 | 19.23v 1891 |
. . . . . . 7
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33 | 23, 31, 32 | 3bitr2i 264 |
. . . . . 6
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34 | 14, 33 | bitr4i 243 |
. . . . 5
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35 | 34 | albii 1566 |
. . . 4
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36 | dfss2 3263 |
. . . 4
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37 | ralcom4 2878 |
. . . 4
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38 | 35, 36, 37 | 3bitr4i 268 |
. . 3
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39 | 38 | biimpri 197 |
. 2
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40 | df-clos1 5874 |
. . . . 5
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41 | 1, 40 | eqtri 2373 |
. . . 4
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42 | sseq2 3294 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
43 | imaeq2 4939 |
. . . . . . . . . 10
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44 | id 19 |
. . . . . . . . . 10
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45 | 43, 44 | sseq12d 3301 |
. . . . . . . . 9
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46 | 42, 45 | anbi12d 691 |
. . . . . . . 8
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47 | 46 | elabg 2987 |
. . . . . . 7
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48 | 47 | biimprd 214 |
. . . . . 6
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49 | 48 | 3impib 1149 |
. . . . 5
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50 | intss1 3942 |
. . . . 5
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51 | 49, 50 | syl 15 |
. . . 4
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52 | 41, 51 | syl5eqss 3316 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | inss1 3476 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
54 | 52, 53 | syl6ss 3285 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 7, 11, 39, 54 | syl3an 1224 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
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 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-1st 4724 df-swap 4725 df-sset 4726 df-co 4727 df-ima 4728 df-si 4729 df-id 4768 df-xp 4785 df-cnv 4786 df-rn 4787 df-dm 4788 df-res 4789 df-2nd 4798 df-txp 5737 df-fix 5741 df-ins2 5751 df-ins3 5753 df-image 5755 df-clos1 5874 |
This theorem is referenced by: clos1is 5882 clos1nrel 5887 clos10 5888 spacind 6288 frecxp 6315 |
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