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Mirrors > Home > MPE Home > Th. List > sossfld | Structured version Visualization version Unicode version |
Description: The base set of a strict
order is contained in the field of the
relation, except possibly for one element (note that
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Ref | Expression |
---|---|
sossfld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4110 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | sotrieq 4804 |
. . . . . . 7
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3 | 2 | necon2abid 2678 |
. . . . . 6
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4 | 3 | anass1rs 821 |
. . . . 5
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5 | breldmg 5062 |
. . . . . . . . . 10
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6 | 5 | 3expia 1217 |
. . . . . . . . 9
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7 | 6 | adantll 725 |
. . . . . . . 8
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8 | 7 | an32s 818 |
. . . . . . 7
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9 | brelrng 5086 |
. . . . . . . . 9
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10 | 9 | 3expia 1217 |
. . . . . . . 8
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11 | 10 | adantll 725 |
. . . . . . 7
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12 | 8, 11 | orim12d 854 |
. . . . . 6
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13 | elun 3586 |
. . . . . 6
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14 | 12, 13 | syl6ibr 235 |
. . . . 5
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15 | 4, 14 | sylbird 243 |
. . . 4
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16 | 15 | expimpd 612 |
. . 3
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17 | 1, 16 | syl5bi 225 |
. 2
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18 | 17 | ssrdv 3450 |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4541 ax-nul 4550 ax-pr 4656 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rab 2758 df-v 3059 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-sn 3981 df-pr 3983 df-op 3987 df-br 4419 df-opab 4478 df-po 4777 df-so 4778 df-cnv 4864 df-dm 4866 df-rn 4867 |
This theorem is referenced by: sofld 5306 soex 6768 |
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