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Mirrors > Home > NFE Home > Th. List > clos1conn | Unicode version |
Description: If a class is connected to an element of a closure via , then it is a member of the closure. Theorem IX.5.14 of [Rosser] p. 246. (Contributed by SF, 13-Feb-2015.) |
Ref | Expression |
---|---|
clos1base.1 | Clos1 |
Ref | Expression |
---|---|
clos1conn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brex 4689 | . . 3 | |
2 | 1 | adantl 452 | . 2 |
3 | eleq1 2413 | . . . . 5 | |
4 | breq1 4642 | . . . . 5 | |
5 | 3, 4 | anbi12d 691 | . . . 4 |
6 | 5 | imbi1d 308 | . . 3 |
7 | breq2 4643 | . . . . 5 | |
8 | 7 | anbi2d 684 | . . . 4 |
9 | eleq1 2413 | . . . 4 | |
10 | 8, 9 | imbi12d 311 | . . 3 |
11 | breq1 4642 | . . . . . . . . . . . . . 14 | |
12 | 11 | rspcev 2955 | . . . . . . . . . . . . 13 |
13 | elima 4754 | . . . . . . . . . . . . 13 | |
14 | 12, 13 | sylibr 203 | . . . . . . . . . . . 12 |
15 | 14 | ancoms 439 | . . . . . . . . . . 11 |
16 | ssel 3267 | . . . . . . . . . . 11 | |
17 | 15, 16 | syl5 28 | . . . . . . . . . 10 |
18 | 17 | exp3a 425 | . . . . . . . . 9 |
19 | 18 | com12 27 | . . . . . . . 8 |
20 | 19 | adantld 453 | . . . . . . 7 |
21 | 20 | a2d 23 | . . . . . 6 |
22 | 21 | alimdv 1621 | . . . . 5 |
23 | clos1base.1 | . . . . . . . 8 Clos1 | |
24 | df-clos1 5873 | . . . . . . . 8 Clos1 | |
25 | 23, 24 | eqtri 2373 | . . . . . . 7 |
26 | 25 | eleq2i 2417 | . . . . . 6 |
27 | vex 2862 | . . . . . . 7 | |
28 | 27 | elintab 3937 | . . . . . 6 |
29 | 26, 28 | bitri 240 | . . . . 5 |
30 | 25 | eleq2i 2417 | . . . . . 6 |
31 | vex 2862 | . . . . . . 7 | |
32 | 31 | elintab 3937 | . . . . . 6 |
33 | 30, 32 | bitri 240 | . . . . 5 |
34 | 22, 29, 33 | 3imtr4g 261 | . . . 4 |
35 | 34 | impcom 419 | . . 3 |
36 | 6, 10, 35 | vtocl2g 2918 | . 2 |
37 | 2, 36 | mpcom 32 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 wal 1540 wceq 1642 wcel 1710 cab 2339 wrex 2615 cvv 2859 wss 3257 cint 3926 class class class wbr 4639 cima 4722 Clos1 cclos1 5872 |
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-br 4640 df-ima 4727 df-clos1 5873 |
This theorem is referenced by: clos1induct 5880 clos1basesuc 5882 spaccl 6286 dmfrec 6316 |
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