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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 3187 | . . . 4 | |
2 | eldif 3050 | . . . . 5 | |
3 | 2 | orbi2i 736 | . . . 4 |
4 | orc 686 | . . . . . . 7 | |
5 | olc 685 | . . . . . . 7 | |
6 | 4, 5 | jca 304 | . . . . . 6 |
7 | olc 685 | . . . . . . 7 | |
8 | orc 686 | . . . . . . 7 | |
9 | 7, 8 | anim12i 336 | . . . . . 6 |
10 | 6, 9 | jaoi 690 | . . . . 5 |
11 | simpl 108 | . . . . . . 7 | |
12 | 11 | orcd 707 | . . . . . 6 |
13 | olc 685 | . . . . . 6 | |
14 | orc 686 | . . . . . . 7 | |
15 | 14 | adantr 274 | . . . . . 6 |
16 | 14 | adantl 275 | . . . . . 6 |
17 | 12, 13, 15, 16 | ccase 933 | . . . . 5 |
18 | 10, 17 | impbii 125 | . . . 4 |
19 | 1, 3, 18 | 3bitri 205 | . . 3 |
20 | elun 3187 | . . . . . 6 | |
21 | 20 | biimpri 132 | . . . . 5 |
22 | pm4.53r 725 | . . . . . 6 | |
23 | eldif 3050 | . . . . . 6 | |
24 | 22, 23 | sylnibr 651 | . . . . 5 |
25 | 21, 24 | anim12i 336 | . . . 4 |
26 | eldif 3050 | . . . 4 | |
27 | 25, 26 | sylibr 133 | . . 3 |
28 | 19, 27 | sylbi 120 | . 2 |
29 | 28 | ssriv 3071 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wo 682 wcel 1465 cdif 3038 cun 3039 wss 3041 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 |
This theorem is referenced by: (None) |
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