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Mirrors > Home > ILE Home > Th. List > unssin | Unicode version |
Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.) |
Ref | Expression |
---|---|
unssin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oranim 840 | . . . . 5 | |
2 | eldifn 3095 | . . . . . 6 | |
3 | eldifn 3095 | . . . . . 6 | |
4 | 2, 3 | anim12i 331 | . . . . 5 |
5 | 1, 4 | nsyl 590 | . . . 4 |
6 | elin 3155 | . . . 4 | |
7 | 5, 6 | sylnibr 634 | . . 3 |
8 | elun 3113 | . . 3 | |
9 | vex 2604 | . . . 4 | |
10 | eldif 2982 | . . . 4 | |
11 | 9, 10 | mpbiran 881 | . . 3 |
12 | 7, 8, 11 | 3imtr4i 199 | . 2 |
13 | 12 | ssriv 3003 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 102 wo 661 wcel 1433 cvv 2601 cdif 2970 cun 2971 cin 2972 wss 2973 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 |
This theorem is referenced by: (None) |
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