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Mirrors > Home > MPE Home > Th. List > cnvps | Structured version Visualization version Unicode version |
Description: The converse of a poset
is a poset. In the general case
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Ref | Expression |
---|---|
cnvps |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5207 |
. . 3
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2 | 1 | a1i 11 |
. 2
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3 | cnvco 5020 |
. . 3
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4 | pstr2 16451 |
. . . 4
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5 | cnvss 5007 |
. . . 4
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6 | 4, 5 | syl 17 |
. . 3
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7 | 3, 6 | syl5eqssr 3477 |
. 2
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8 | psrel 16449 |
. . . . . 6
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9 | dfrel2 5286 |
. . . . . 6
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10 | 8, 9 | sylib 200 |
. . . . 5
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11 | 10 | ineq2d 3634 |
. . . 4
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12 | incom 3625 |
. . . 4
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13 | 11, 12 | syl6eq 2501 |
. . 3
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14 | psref2 16450 |
. . 3
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15 | relcnvfld 5367 |
. . . . 5
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16 | 8, 15 | syl 17 |
. . . 4
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17 | 16 | reseq2d 5105 |
. . 3
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18 | 13, 14, 17 | 3eqtrd 2489 |
. 2
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19 | cnvexg 6739 |
. . 3
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20 | isps 16448 |
. . 3
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21 | 19, 20 | syl 17 |
. 2
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22 | 2, 7, 18, 21 | mpbir3and 1191 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ps 16446 |
This theorem is referenced by: cnvpsb 16459 cnvtsr 16468 ordtcnv 20217 xrge0iifhmeo 28742 |
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