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Mirrors > Home > ILE Home > Th. List > triun | Unicode version |
Description: The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
triun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 3817 | . . . 4 | |
2 | r19.29 2569 | . . . . 5 | |
3 | nfcv 2281 | . . . . . . 7 | |
4 | nfiu1 3843 | . . . . . . 7 | |
5 | 3, 4 | nfss 3090 | . . . . . 6 |
6 | trss 4035 | . . . . . . . 8 | |
7 | 6 | imp 123 | . . . . . . 7 |
8 | ssiun2 3856 | . . . . . . . 8 | |
9 | sstr2 3104 | . . . . . . . 8 | |
10 | 8, 9 | syl5com 29 | . . . . . . 7 |
11 | 7, 10 | syl5 32 | . . . . . 6 |
12 | 5, 11 | rexlimi 2542 | . . . . 5 |
13 | 2, 12 | syl 14 | . . . 4 |
14 | 1, 13 | sylan2b 285 | . . 3 |
15 | 14 | ralrimiva 2505 | . 2 |
16 | dftr3 4030 | . 2 | |
17 | 15, 16 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 wral 2416 wrex 2417 wss 3071 ciun 3813 wtr 4026 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-in 3077 df-ss 3084 df-uni 3737 df-iun 3815 df-tr 4027 |
This theorem is referenced by: truni 4040 |
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