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Mirrors > Home > ILE Home > Th. List > trin | Unicode version |
Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.) |
Ref | Expression |
---|---|
trin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3310 | . . . . 5 | |
2 | trss 4094 | . . . . . 6 | |
3 | trss 4094 | . . . . . 6 | |
4 | 2, 3 | im2anan9 593 | . . . . 5 |
5 | 1, 4 | syl5bi 151 | . . . 4 |
6 | ssin 3349 | . . . 4 | |
7 | 5, 6 | syl6ib 160 | . . 3 |
8 | 7 | ralrimiv 2542 | . 2 |
9 | dftr3 4089 | . 2 | |
10 | 8, 9 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2141 wral 2448 cin 3120 wss 3121 wtr 4085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-in 3127 df-ss 3134 df-uni 3795 df-tr 4086 |
This theorem is referenced by: ordin 4368 |
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