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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 6702 | . 2 | |
2 | vex 2724 | . . . 4 | |
3 | 2 | brdom 6707 | . . 3 |
4 | vex 2724 | . . . 4 | |
5 | 4 | brdom 6707 | . . 3 |
6 | eeanv 1919 | . . . 4 | |
7 | f1co 5399 | . . . . . . . 8 | |
8 | 7 | ancoms 266 | . . . . . . 7 |
9 | vex 2724 | . . . . . . . . 9 | |
10 | vex 2724 | . . . . . . . . 9 | |
11 | 9, 10 | coex 5143 | . . . . . . . 8 |
12 | f1eq1 5382 | . . . . . . . 8 | |
13 | 11, 12 | spcev 2816 | . . . . . . 7 |
14 | 8, 13 | syl 14 | . . . . . 6 |
15 | 4 | brdom 6707 | . . . . . 6 |
16 | 14, 15 | sylibr 133 | . . . . 5 |
17 | 16 | exlimivv 1883 | . . . 4 |
18 | 6, 17 | sylbir 134 | . . 3 |
19 | 3, 5, 18 | syl2anb 289 | . 2 |
20 | 1, 19 | vtoclr 4646 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wex 1479 class class class wbr 3976 ccom 4602 wf1 5179 cdom 6696 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-dom 6699 |
This theorem is referenced by: endomtr 6747 domentr 6748 cnvct 6766 ssct 6775 nndomo 6821 infnfi 6852 xpct 12272 |
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