Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > poinxp | Unicode version |
Description: Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.) |
Ref | Expression |
---|---|
poinxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 518 | . . . . . . . 8 | |
2 | brinxp 4602 | . . . . . . . 8 | |
3 | 1, 1, 2 | syl2anc 408 | . . . . . . 7 |
4 | 3 | notbid 656 | . . . . . 6 |
5 | brinxp 4602 | . . . . . . . . 9 | |
6 | 5 | adantr 274 | . . . . . . . 8 |
7 | brinxp 4602 | . . . . . . . . 9 | |
8 | 7 | adantll 467 | . . . . . . . 8 |
9 | 6, 8 | anbi12d 464 | . . . . . . 7 |
10 | brinxp 4602 | . . . . . . . 8 | |
11 | 10 | adantlr 468 | . . . . . . 7 |
12 | 9, 11 | imbi12d 233 | . . . . . 6 |
13 | 4, 12 | anbi12d 464 | . . . . 5 |
14 | 13 | ralbidva 2431 | . . . 4 |
15 | 14 | ralbidva 2431 | . . 3 |
16 | 15 | ralbiia 2447 | . 2 |
17 | df-po 4213 | . 2 | |
18 | df-po 4213 | . 2 | |
19 | 16, 17, 18 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 1480 wral 2414 cin 3065 class class class wbr 3924 wpo 4211 cxp 4532 |
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-in1 603 ax-in2 604 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-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-po 4213 df-xp 4540 |
This theorem is referenced by: soinxp 4604 |
Copyright terms: Public domain | W3C validator |