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 807 | . . . . 5 | |
2 | eldifn 3067 | . . . . . 6 | |
3 | eldifn 3067 | . . . . . 6 | |
4 | 2, 3 | anim12i 321 | . . . . 5 |
5 | 1, 4 | nsyl 558 | . . . 4 |
6 | elin 3126 | . . . 4 | |
7 | 5, 6 | sylnibr 602 | . . 3 |
8 | elun 3084 | . . 3 | |
9 | vex 2560 | . . . 4 | |
10 | eldif 2927 | . . . 4 | |
11 | 9, 10 | mpbiran 847 | . . 3 |
12 | 7, 8, 11 | 3imtr4i 190 | . 2 |
13 | 12 | ssriv 2949 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 97 wo 629 wcel 1393 cvv 2557 cdif 2914 cun 2915 cin 2916 wss 2917 |
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-in1 544 ax-in2 545 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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |