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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 496 |
. . . . . . . 8
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2 | brinxp 4506 |
. . . . . . . 8
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3 | 1, 1, 2 | syl2anc 403 |
. . . . . . 7
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4 | 3 | notbid 627 |
. . . . . 6
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5 | brinxp 4506 |
. . . . . . . . 9
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6 | 5 | adantr 270 |
. . . . . . . 8
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7 | brinxp 4506 |
. . . . . . . . 9
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8 | 7 | adantll 460 |
. . . . . . . 8
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9 | 6, 8 | anbi12d 457 |
. . . . . . 7
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10 | brinxp 4506 |
. . . . . . . 8
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11 | 10 | adantlr 461 |
. . . . . . 7
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12 | 9, 11 | imbi12d 232 |
. . . . . 6
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13 | 4, 12 | anbi12d 457 |
. . . . 5
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14 | 13 | ralbidva 2376 |
. . . 4
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15 | 14 | ralbidva 2376 |
. . 3
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16 | 15 | ralbiia 2392 |
. 2
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17 | df-po 4123 |
. 2
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18 | df-po 4123 |
. 2
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19 | 16, 17, 18 | 3bitr4i 210 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-po 4123 df-xp 4444 |
This theorem is referenced by: soinxp 4508 |
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