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Mirrors > Home > NFE Home > Th. List > elxp4 | Unicode version |
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. (Contributed by set.mm contributors, 17-Feb-2004.) |
Ref | Expression |
---|---|
elxp4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4801 | . 2 | |
2 | sneq 3744 | . . . . . . . . . . 11 | |
3 | 2 | rneqd 4958 | . . . . . . . . . 10 |
4 | 3 | unieqd 3902 | . . . . . . . . 9 |
5 | vex 2862 | . . . . . . . . . 10 | |
6 | vex 2862 | . . . . . . . . . 10 | |
7 | 5, 6 | op2nda 5076 | . . . . . . . . 9 |
8 | 4, 7 | syl6req 2402 | . . . . . . . 8 |
9 | 8 | adantr 451 | . . . . . . 7 |
10 | 9 | pm4.71ri 614 | . . . . . 6 |
11 | 10 | exbii 1582 | . . . . 5 |
12 | snex 4111 | . . . . . . . 8 | |
13 | 12 | rnex 5107 | . . . . . . 7 |
14 | 13 | uniex 4317 | . . . . . 6 |
15 | opeq2 4579 | . . . . . . . 8 | |
16 | 15 | eqeq2d 2364 | . . . . . . 7 |
17 | eleq1 2413 | . . . . . . . 8 | |
18 | 17 | anbi2d 684 | . . . . . . 7 |
19 | 16, 18 | anbi12d 691 | . . . . . 6 |
20 | 14, 19 | ceqsexv 2894 | . . . . 5 |
21 | 11, 20 | bitri 240 | . . . 4 |
22 | sneq 3744 | . . . . . . . . 9 | |
23 | 22 | dmeqd 4909 | . . . . . . . 8 |
24 | 23 | unieqd 3902 | . . . . . . 7 |
25 | 5, 14 | op1sta 5072 | . . . . . . 7 |
26 | 24, 25 | syl6req 2402 | . . . . . 6 |
27 | 26 | pm4.71ri 614 | . . . . 5 |
28 | 27 | anbi1i 676 | . . . 4 |
29 | anass 630 | . . . 4 | |
30 | 21, 28, 29 | 3bitri 262 | . . 3 |
31 | 30 | exbii 1582 | . 2 |
32 | 12 | dmex 5106 | . . . 4 |
33 | 32 | uniex 4317 | . . 3 |
34 | opeq1 4578 | . . . . 5 | |
35 | 34 | eqeq2d 2364 | . . . 4 |
36 | eleq1 2413 | . . . . 5 | |
37 | 36 | anbi1d 685 | . . . 4 |
38 | 35, 37 | anbi12d 691 | . . 3 |
39 | 33, 38 | ceqsexv 2894 | . 2 |
40 | 1, 31, 39 | 3bitri 262 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 wex 1541 wceq 1642 wcel 1710 csn 3737 cuni 3891 cop 4561 cxp 4770 cdm 4772 crn 4773 |
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 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-swap 4724 df-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 |
This theorem is referenced by: (None) |
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