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Mirrors > Home > NFE Home > Th. List > xpkvexg | Unicode version |
Description: The Kuratowski cross product of with a set is a set. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
xpkvexg | k |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpkeq2 4199 | . . 3 k k | |
2 | 1 | eleq1d 2419 | . 2 k k |
3 | ax-xp 4079 | . . 3 | |
4 | isset 2863 | . . . 4 k k | |
5 | dfcleq 2347 | . . . . . 6 k k | |
6 | elxpk 4196 | . . . . . . . . 9 k | |
7 | vex 2862 | . . . . . . . . . . . 12 | |
8 | 7 | biantrur 492 | . . . . . . . . . . 11 |
9 | 8 | anbi2i 675 | . . . . . . . . . 10 |
10 | 9 | 2exbii 1583 | . . . . . . . . 9 |
11 | 6, 10 | bitr4i 243 | . . . . . . . 8 k |
12 | 11 | bibi2i 304 | . . . . . . 7 k |
13 | 12 | albii 1566 | . . . . . 6 k |
14 | 5, 13 | bitri 240 | . . . . 5 k |
15 | 14 | exbii 1582 | . . . 4 k |
16 | 4, 15 | bitri 240 | . . 3 k |
17 | 3, 16 | mpbir 200 | . 2 k |
18 | 2, 17 | vtoclg 2914 | 1 k |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wal 1540 wex 1541 wceq 1642 wcel 1710 cvv 2859 copk 4057 k cxpk 4174 |
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-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-xpk 4185 |
This theorem is referenced by: cnvkexg 4286 xpkexg 4288 ssetkex 4294 |
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