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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 519 | . . . . . . . 8 | |
2 | brinxp 4667 | . . . . . . . 8 | |
3 | 1, 1, 2 | syl2anc 409 | . . . . . . 7 |
4 | 3 | notbid 657 | . . . . . 6 |
5 | brinxp 4667 | . . . . . . . . 9 | |
6 | 5 | adantr 274 | . . . . . . . 8 |
7 | brinxp 4667 | . . . . . . . . 9 | |
8 | 7 | adantll 468 | . . . . . . . 8 |
9 | 6, 8 | anbi12d 465 | . . . . . . 7 |
10 | brinxp 4667 | . . . . . . . 8 | |
11 | 10 | adantlr 469 | . . . . . . 7 |
12 | 9, 11 | imbi12d 233 | . . . . . 6 |
13 | 4, 12 | anbi12d 465 | . . . . 5 |
14 | 13 | ralbidva 2460 | . . . 4 |
15 | 14 | ralbidva 2460 | . . 3 |
16 | 15 | ralbiia 2478 | . 2 |
17 | df-po 4269 | . 2 | |
18 | df-po 4269 | . 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 2135 wral 2442 cin 3111 class class class wbr 3977 wpo 4267 cxp 4597 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-br 3978 df-opab 4039 df-po 4269 df-xp 4605 |
This theorem is referenced by: soinxp 4669 |
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