| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > ILE Home > Th. List > undif3ss | Unicode version | ||
| Description: A subset relationship involving class union and class difference. In classical logic, this would be equality rather than subset, as in the first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Jim Kingdon, 28-Jul-2018.) |
| Ref | Expression |
|---|---|
| undif3ss |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elun 3313 |
. . . 4
| |
| 2 | eldif 3174 |
. . . . 5
| |
| 3 | 2 | orbi2i 763 |
. . . 4
|
| 4 | orc 713 |
. . . . . . 7
| |
| 5 | olc 712 |
. . . . . . 7
| |
| 6 | 4, 5 | jca 306 |
. . . . . 6
|
| 7 | olc 712 |
. . . . . . 7
| |
| 8 | orc 713 |
. . . . . . 7
| |
| 9 | 7, 8 | anim12i 338 |
. . . . . 6
|
| 10 | 6, 9 | jaoi 717 |
. . . . 5
|
| 11 | simpl 109 |
. . . . . . 7
| |
| 12 | 11 | orcd 734 |
. . . . . 6
|
| 13 | olc 712 |
. . . . . 6
| |
| 14 | orc 713 |
. . . . . . 7
| |
| 15 | 14 | adantr 276 |
. . . . . 6
|
| 16 | 14 | adantl 277 |
. . . . . 6
|
| 17 | 12, 13, 15, 16 | ccase 966 |
. . . . 5
|
| 18 | 10, 17 | impbii 126 |
. . . 4
|
| 19 | 1, 3, 18 | 3bitri 206 |
. . 3
|
| 20 | elun 3313 |
. . . . . 6
| |
| 21 | 20 | biimpri 133 |
. . . . 5
|
| 22 | pm4.53r 752 |
. . . . . 6
| |
| 23 | eldif 3174 |
. . . . . 6
| |
| 24 | 22, 23 | sylnibr 678 |
. . . . 5
|
| 25 | 21, 24 | anim12i 338 |
. . . 4
|
| 26 | eldif 3174 |
. . . 4
| |
| 27 | 25, 26 | sylibr 134 |
. . 3
|
| 28 | 19, 27 | sylbi 121 |
. 2
|
| 29 | 28 | ssriv 3196 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-v 2773 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |