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Mirrors > Home > NFE Home > Th. List > opeliunxp | Unicode version |
Description: Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.) |
Ref | Expression |
---|---|
opeliunxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2867 | . . 3 | |
2 | opexb 4603 | . . . 4 | |
3 | 2 | simprbi 450 | . . 3 |
4 | 1, 3 | syl 15 | . 2 |
5 | elex 2867 | . . 3 | |
6 | 5 | adantl 452 | . 2 |
7 | vex 2862 | . . . . 5 | |
8 | opexg 4587 | . . . . 5 | |
9 | 7, 8 | mpan 651 | . . . 4 |
10 | df-rex 2620 | . . . . . . 7 | |
11 | nfv 1619 | . . . . . . . 8 | |
12 | nfs1v 2106 | . . . . . . . . 9 | |
13 | nfcv 2489 | . . . . . . . . . . 11 | |
14 | nfcsb1v 3168 | . . . . . . . . . . 11 | |
15 | 13, 14 | nfxp 4810 | . . . . . . . . . 10 |
16 | 15 | nfcri 2483 | . . . . . . . . 9 |
17 | 12, 16 | nfan 1824 | . . . . . . . 8 |
18 | sbequ12 1919 | . . . . . . . . 9 | |
19 | sneq 3744 | . . . . . . . . . . 11 | |
20 | csbeq1a 3144 | . . . . . . . . . . 11 | |
21 | 19, 20 | xpeq12d 4809 | . . . . . . . . . 10 |
22 | 21 | eleq2d 2420 | . . . . . . . . 9 |
23 | 18, 22 | anbi12d 691 | . . . . . . . 8 |
24 | 11, 17, 23 | cbvex 1985 | . . . . . . 7 |
25 | 10, 24 | bitri 240 | . . . . . 6 |
26 | eleq1 2413 | . . . . . . . 8 | |
27 | 26 | anbi2d 684 | . . . . . . 7 |
28 | 27 | exbidv 1626 | . . . . . 6 |
29 | 25, 28 | syl5bb 248 | . . . . 5 |
30 | df-iun 3971 | . . . . 5 | |
31 | 29, 30 | elab2g 2987 | . . . 4 |
32 | 9, 31 | syl 15 | . . 3 |
33 | opelxp 4811 | . . . . . . 7 | |
34 | 33 | anbi2i 675 | . . . . . 6 |
35 | an12 772 | . . . . . 6 | |
36 | elsn 3748 | . . . . . . . 8 | |
37 | equcom 1680 | . . . . . . . 8 | |
38 | 36, 37 | bitri 240 | . . . . . . 7 |
39 | 38 | anbi1i 676 | . . . . . 6 |
40 | 34, 35, 39 | 3bitri 262 | . . . . 5 |
41 | 40 | exbii 1582 | . . . 4 |
42 | sbequ12r 1920 | . . . . . 6 | |
43 | 20 | equcoms 1681 | . . . . . . . 8 |
44 | 43 | eqcomd 2358 | . . . . . . 7 |
45 | 44 | eleq2d 2420 | . . . . . 6 |
46 | 42, 45 | anbi12d 691 | . . . . 5 |
47 | 7, 46 | ceqsexv 2894 | . . . 4 |
48 | 41, 47 | bitri 240 | . . 3 |
49 | 32, 48 | syl6bb 252 | . 2 |
50 | 4, 6, 49 | pm5.21nii 342 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 wex 1541 wceq 1642 wsb 1648 wcel 1710 wrex 2615 cvv 2859 csb 3136 csn 3737 ciun 3969 cop 4561 cxp 4770 |
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-csb 3137 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-iun 3971 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-xp 4784 |
This theorem is referenced by: eliunxp 4821 opeliunxp2 4822 |
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