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Mirrors > Home > ILE Home > Th. List > domtr | Unicode version |
Description: Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
domtr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 6226 | . 2 | |
2 | vex 2560 | . . . 4 | |
3 | 2 | brdom 6231 | . . 3 |
4 | vex 2560 | . . . 4 | |
5 | 4 | brdom 6231 | . . 3 |
6 | eeanv 1807 | . . . 4 | |
7 | f1co 5101 | . . . . . . . 8 | |
8 | 7 | ancoms 255 | . . . . . . 7 |
9 | vex 2560 | . . . . . . . . 9 | |
10 | vex 2560 | . . . . . . . . 9 | |
11 | 9, 10 | coex 4863 | . . . . . . . 8 |
12 | f1eq1 5087 | . . . . . . . 8 | |
13 | 11, 12 | spcev 2647 | . . . . . . 7 |
14 | 8, 13 | syl 14 | . . . . . 6 |
15 | 4 | brdom 6231 | . . . . . 6 |
16 | 14, 15 | sylibr 137 | . . . . 5 |
17 | 16 | exlimivv 1776 | . . . 4 |
18 | 6, 17 | sylbir 125 | . . 3 |
19 | 3, 5, 18 | syl2anb 275 | . 2 |
20 | 1, 19 | vtoclr 4388 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wex 1381 class class class wbr 3764 ccom 4349 wf1 4899 cdom 6220 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-dom 6223 |
This theorem is referenced by: endomtr 6270 domentr 6271 nndomo 6326 |
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