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Mirrors > Home > MPE Home > Th. List > poinxp | Structured version Visualization version Unicode version |
Description: Intersection of partial order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.) |
Ref | Expression |
---|---|
poinxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brinxp 5181 | . . . . . . . . 9 | |
2 | 1 | anidms 677 | . . . . . . . 8 |
3 | 2 | ad2antrr 762 | . . . . . . 7 |
4 | 3 | notbid 308 | . . . . . 6 |
5 | brinxp 5181 | . . . . . . . . 9 | |
6 | 5 | adantr 481 | . . . . . . . 8 |
7 | brinxp 5181 | . . . . . . . . 9 | |
8 | 7 | adantll 750 | . . . . . . . 8 |
9 | 6, 8 | anbi12d 747 | . . . . . . 7 |
10 | brinxp 5181 | . . . . . . . 8 | |
11 | 10 | adantlr 751 | . . . . . . 7 |
12 | 9, 11 | imbi12d 334 | . . . . . 6 |
13 | 4, 12 | anbi12d 747 | . . . . 5 |
14 | 13 | ralbidva 2985 | . . . 4 |
15 | 14 | ralbidva 2985 | . . 3 |
16 | 15 | ralbiia 2979 | . 2 |
17 | df-po 5035 | . 2 | |
18 | df-po 5035 | . 2 | |
19 | 16, 17, 18 | 3bitr4i 292 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wcel 1990 wral 2912 cin 3573 class class class wbr 4653 wpo 5033 cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-po 5035 df-xp 5120 |
This theorem is referenced by: soinxp 5183 |
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