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Mirrors > Home > MPE Home > Th. List > trin | Structured version Visualization version Unicode version |
Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.) |
Ref | Expression |
---|---|
trin |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3608 |
. . . . 5
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2 | trss 4499 |
. . . . . 6
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3 | trss 4499 |
. . . . . 6
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4 | 2, 3 | im2anan9 853 |
. . . . 5
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5 | 1, 4 | syl5bi 225 |
. . . 4
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6 | ssin 3645 |
. . . 4
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7 | 5, 6 | syl6ib 234 |
. . 3
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8 | 7 | ralrimiv 2808 |
. 2
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9 | dftr3 4494 |
. 2
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10 | 8, 9 | sylibr 217 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 |
This theorem depends on definitions: df-bi 190 df-an 378 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ral 2761 df-v 3033 df-in 3397 df-ss 3404 df-uni 4191 df-tr 4491 |
This theorem is referenced by: ordin 5460 tcmin 8243 ingru 9258 gruina 9261 dfon2lem4 30503 |
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