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Mirrors > Home > ILE Home > Th. List > xpidtr | Unicode version |
Description: A square cross product is a transitive relation. (Contributed by FL, 31-Jul-2009.) |
Ref | Expression |
---|---|
xpidtr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brxp 4565 | . . . . . 6 | |
2 | brxp 4565 | . . . . . . . . 9 | |
3 | brxp 4565 | . . . . . . . . . . 11 | |
4 | 3 | simplbi2com 1420 | . . . . . . . . . 10 |
5 | 4 | adantl 275 | . . . . . . . . 9 |
6 | 2, 5 | sylbi 120 | . . . . . . . 8 |
7 | 6 | com12 30 | . . . . . . 7 |
8 | 7 | adantr 274 | . . . . . 6 |
9 | 1, 8 | sylbi 120 | . . . . 5 |
10 | 9 | imp 123 | . . . 4 |
11 | 10 | ax-gen 1425 | . . 3 |
12 | 11 | gen2 1426 | . 2 |
13 | cotr 4915 | . 2 | |
14 | 12, 13 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wcel 1480 wss 3066 class class class wbr 3924 cxp 4532 ccom 4538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-co 4543 |
This theorem is referenced by: trinxp 4927 xpider 6493 |
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