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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 3875 | . . . 4 | |
2 | r19.29 2607 | . . . . 5 | |
3 | nfcv 2312 | . . . . . . 7 | |
4 | nfiu1 3901 | . . . . . . 7 | |
5 | 3, 4 | nfss 3140 | . . . . . 6 |
6 | trss 4094 | . . . . . . . 8 | |
7 | 6 | imp 123 | . . . . . . 7 |
8 | ssiun2 3914 | . . . . . . . 8 | |
9 | sstr2 3154 | . . . . . . . 8 | |
10 | 8, 9 | syl5com 29 | . . . . . . 7 |
11 | 7, 10 | syl5 32 | . . . . . 6 |
12 | 5, 11 | rexlimi 2580 | . . . . 5 |
13 | 2, 12 | syl 14 | . . . 4 |
14 | 1, 13 | sylan2b 285 | . . 3 |
15 | 14 | ralrimiva 2543 | . 2 |
16 | dftr3 4089 | . 2 | |
17 | 15, 16 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2141 wral 2448 wrex 2449 wss 3121 ciun 3871 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-rex 2454 df-v 2732 df-in 3127 df-ss 3134 df-uni 3795 df-iun 3873 df-tr 4086 |
This theorem is referenced by: truni 4099 |
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