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Mirrors > Home > MPE Home > Th. List > elxp4 | Structured version Visualization version Unicode version |
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7111, elxp6 7200, and elxp7 7201. (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
elxp4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 5131 | . 2 | |
2 | sneq 4187 | . . . . . . . . . . . 12 | |
3 | 2 | rneqd 5353 | . . . . . . . . . . 11 |
4 | 3 | unieqd 4446 | . . . . . . . . . 10 |
5 | vex 3203 | . . . . . . . . . . 11 | |
6 | vex 3203 | . . . . . . . . . . 11 | |
7 | 5, 6 | op2nda 5620 | . . . . . . . . . 10 |
8 | 4, 7 | syl6req 2673 | . . . . . . . . 9 |
9 | 8 | pm4.71ri 665 | . . . . . . . 8 |
10 | 9 | anbi1i 731 | . . . . . . 7 |
11 | anass 681 | . . . . . . 7 | |
12 | 10, 11 | bitri 264 | . . . . . 6 |
13 | 12 | exbii 1774 | . . . . 5 |
14 | snex 4908 | . . . . . . . 8 | |
15 | 14 | rnex 7100 | . . . . . . 7 |
16 | 15 | uniex 6953 | . . . . . 6 |
17 | opeq2 4403 | . . . . . . . 8 | |
18 | 17 | eqeq2d 2632 | . . . . . . 7 |
19 | eleq1 2689 | . . . . . . . 8 | |
20 | 19 | anbi2d 740 | . . . . . . 7 |
21 | 18, 20 | anbi12d 747 | . . . . . 6 |
22 | 16, 21 | ceqsexv 3242 | . . . . 5 |
23 | 13, 22 | bitri 264 | . . . 4 |
24 | sneq 4187 | . . . . . . . . 9 | |
25 | 24 | dmeqd 5326 | . . . . . . . 8 |
26 | 25 | unieqd 4446 | . . . . . . 7 |
27 | 5, 16 | op1sta 5617 | . . . . . . 7 |
28 | 26, 27 | syl6req 2673 | . . . . . 6 |
29 | 28 | pm4.71ri 665 | . . . . 5 |
30 | 29 | anbi1i 731 | . . . 4 |
31 | anass 681 | . . . 4 | |
32 | 23, 30, 31 | 3bitri 286 | . . 3 |
33 | 32 | exbii 1774 | . 2 |
34 | 14 | dmex 7099 | . . . 4 |
35 | 34 | uniex 6953 | . . 3 |
36 | opeq1 4402 | . . . . 5 | |
37 | 36 | eqeq2d 2632 | . . . 4 |
38 | eleq1 2689 | . . . . 5 | |
39 | 38 | anbi1d 741 | . . . 4 |
40 | 37, 39 | anbi12d 747 | . . 3 |
41 | 35, 40 | ceqsexv 3242 | . 2 |
42 | 1, 33, 41 | 3bitri 286 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 csn 4177 cop 4183 cuni 4436 cxp 5112 cdm 5114 crn 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: elxp6 7200 xpdom2 8055 |
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