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Mirrors > Home > MPE Home > Th. List > unielxp | Structured version Visualization version Unicode version |
Description: The membership relation for a Cartesian product is inherited by union. (Contributed by NM, 16-Sep-2006.) |
Ref | Expression |
---|---|
unielxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp7 7201 | . 2 | |
2 | elvvuni 5179 | . . . 4 | |
3 | 2 | adantr 481 | . . 3 |
4 | simprl 794 | . . . . . 6 | |
5 | eleq2 2690 | . . . . . . . 8 | |
6 | eleq1 2689 | . . . . . . . . 9 | |
7 | fveq2 6191 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2686 | . . . . . . . . . 10 |
9 | fveq2 6191 | . . . . . . . . . . 11 | |
10 | 9 | eleq1d 2686 | . . . . . . . . . 10 |
11 | 8, 10 | anbi12d 747 | . . . . . . . . 9 |
12 | 6, 11 | anbi12d 747 | . . . . . . . 8 |
13 | 5, 12 | anbi12d 747 | . . . . . . 7 |
14 | 13 | spcegv 3294 | . . . . . 6 |
15 | 4, 14 | mpcom 38 | . . . . 5 |
16 | eluniab 4447 | . . . . 5 | |
17 | 15, 16 | sylibr 224 | . . . 4 |
18 | xp2 7203 | . . . . . 6 | |
19 | df-rab 2921 | . . . . . 6 | |
20 | 18, 19 | eqtri 2644 | . . . . 5 |
21 | 20 | unieqi 4445 | . . . 4 |
22 | 17, 21 | syl6eleqr 2712 | . . 3 |
23 | 3, 22 | mpancom 703 | . 2 |
24 | 1, 23 | sylbi 207 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wex 1704 wcel 1990 cab 2608 crab 2916 cvv 3200 cuni 4436 cxp 5112 cfv 5888 c1st 7166 c2nd 7167 |
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-pow 4843 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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: (None) |
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