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Mirrors > Home > MPE Home > Th. List > sotri3 | Structured version Visualization version Unicode version |
Description: A transitivity relation.
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Ref | Expression |
---|---|
soi.1 |
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soi.2 |
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Ref | Expression |
---|---|
sotri3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | soi.2 |
. . . . . 6
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2 | 1 | brel 4886 |
. . . . 5
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3 | 2 | simprd 465 |
. . . 4
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4 | soi.1 |
. . . . . . . 8
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5 | sotric 4784 |
. . . . . . . 8
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6 | 4, 5 | mpan 677 |
. . . . . . 7
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7 | 6 | con2bid 331 |
. . . . . 6
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8 | breq2 4409 |
. . . . . . . 8
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9 | 8 | biimprd 227 |
. . . . . . 7
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10 | 4, 1 | sotri 5230 |
. . . . . . . 8
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11 | 10 | expcom 437 |
. . . . . . 7
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12 | 9, 11 | jaoi 381 |
. . . . . 6
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13 | 7, 12 | syl6bir 233 |
. . . . 5
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14 | 13 | com3r 82 |
. . . 4
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15 | 3, 14 | mpan2d 681 |
. . 3
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16 | 15 | com12 32 |
. 2
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17 | 16 | 3imp 1203 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pr 4642 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-op 3977 df-br 4406 df-opab 4465 df-po 4758 df-so 4759 df-xp 4843 |
This theorem is referenced by: archnq 9410 |
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