Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eliunxp | Structured version Visualization version Unicode version |
Description: Membership in a union of Cartesian products. Analogue of elxp 5131 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
eliunxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5227 | . . . . . 6 | |
2 | 1 | rgenw 2924 | . . . . 5 |
3 | reliun 5239 | . . . . 5 | |
4 | 2, 3 | mpbir 221 | . . . 4 |
5 | elrel 5222 | . . . 4 | |
6 | 4, 5 | mpan 706 | . . 3 |
7 | 6 | pm4.71ri 665 | . 2 |
8 | nfiu1 4550 | . . . 4 | |
9 | 8 | nfel2 2781 | . . 3 |
10 | 9 | 19.41 2103 | . 2 |
11 | 19.41v 1914 | . . . 4 | |
12 | eleq1 2689 | . . . . . . 7 | |
13 | opeliunxp 5170 | . . . . . . 7 | |
14 | 12, 13 | syl6bb 276 | . . . . . 6 |
15 | 14 | pm5.32i 669 | . . . . 5 |
16 | 15 | exbii 1774 | . . . 4 |
17 | 11, 16 | bitr3i 266 | . . 3 |
18 | 17 | exbii 1774 | . 2 |
19 | 7, 10, 18 | 3bitr2i 288 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 wral 2912 csn 4177 cop 4183 ciun 4520 cxp 5112 wrel 5119 |
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-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 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-iun 4522 df-opab 4713 df-xp 5120 df-rel 5121 |
This theorem is referenced by: raliunxp 5261 dfmpt3 6014 mpt2mptx 6751 fsumcom2 14505 fsumcom2OLD 14506 fprodcom2 14714 fprodcom2OLD 14715 isfunc 16524 gsum2d2 18373 dprd2d2 18443 fsumvma 24938 mpt2mptxf 29477 poimirlem26 33435 dvnprodlem1 40161 |
Copyright terms: Public domain | W3C validator |