Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 771 | . . . . 5 | |
2 | eldifn 3231 | . . . . . 6 | |
3 | eldifn 3231 | . . . . . 6 | |
4 | 2, 3 | anim12i 336 | . . . . 5 |
5 | 1, 4 | nsyl 618 | . . . 4 |
6 | elin 3291 | . . . 4 | |
7 | 5, 6 | sylnibr 667 | . . 3 |
8 | elun 3249 | . . 3 | |
9 | vex 2715 | . . . 4 | |
10 | eldif 3111 | . . . 4 | |
11 | 9, 10 | mpbiran 925 | . . 3 |
12 | 7, 8, 11 | 3imtr4i 200 | . 2 |
13 | 12 | ssriv 3132 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wo 698 wcel 2128 cvv 2712 cdif 3099 cun 3100 cin 3101 wss 3102 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |