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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 3113 | . . . 4 | |
2 | eldif 2982 | . . . . 5 | |
3 | 2 | orbi2i 711 | . . . 4 |
4 | orc 665 | . . . . . . 7 | |
5 | olc 664 | . . . . . . 7 | |
6 | 4, 5 | jca 300 | . . . . . 6 |
7 | olc 664 | . . . . . . 7 | |
8 | orc 665 | . . . . . . 7 | |
9 | 7, 8 | anim12i 331 | . . . . . 6 |
10 | 6, 9 | jaoi 668 | . . . . 5 |
11 | simpl 107 | . . . . . . 7 | |
12 | 11 | orcd 684 | . . . . . 6 |
13 | olc 664 | . . . . . 6 | |
14 | orc 665 | . . . . . . 7 | |
15 | 14 | adantr 270 | . . . . . 6 |
16 | 14 | adantl 271 | . . . . . 6 |
17 | 12, 13, 15, 16 | ccase 905 | . . . . 5 |
18 | 10, 17 | impbii 124 | . . . 4 |
19 | 1, 3, 18 | 3bitri 204 | . . 3 |
20 | elun 3113 | . . . . . 6 | |
21 | 20 | biimpri 131 | . . . . 5 |
22 | pm4.53r 837 | . . . . . 6 | |
23 | eldif 2982 | . . . . . 6 | |
24 | 22, 23 | sylnibr 634 | . . . . 5 |
25 | 21, 24 | anim12i 331 | . . . 4 |
26 | eldif 2982 | . . . 4 | |
27 | 25, 26 | sylibr 132 | . . 3 |
28 | 19, 27 | sylbi 119 | . 2 |
29 | 28 | ssriv 3003 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 102 wo 661 wcel 1433 cdif 2970 cun 2971 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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