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Mirrors > Home > MPE Home > Th. List > difin0ss | Unicode version |
Description: Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) |
Ref | Expression |
---|---|
difin0ss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3800 | . 2 | |
2 | iman 424 | . . . . . 6 | |
3 | elin 3686 | . . . . . . . 8 | |
4 | eldif 3485 | . . . . . . . . 9 | |
5 | 4 | anbi1i 695 | . . . . . . . 8 |
6 | 3, 5 | bitri 249 | . . . . . . 7 |
7 | ancom 450 | . . . . . . 7 | |
8 | annim 425 | . . . . . . . 8 | |
9 | 8 | anbi2i 694 | . . . . . . 7 |
10 | 6, 7, 9 | 3bitr2i 273 | . . . . . 6 |
11 | 2, 10 | xchbinxr 311 | . . . . 5 |
12 | ax-2 7 | . . . . 5 | |
13 | 11, 12 | sylbir 213 | . . . 4 |
14 | 13 | al2imi 1636 | . . 3 |
15 | dfss2 3492 | . . 3 | |
16 | dfss2 3492 | . . 3 | |
17 | 14, 15, 16 | 3imtr4g 270 | . 2 |
18 | 1, 17 | sylbi 195 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 -> wi 4
/\ wa 369 A. wal 1393 = wceq 1395
e. wcel 1818 \ cdif 3472 i^i cin 3474
C_ wss 3475 c0 3784 |
This theorem is referenced by: tz7.7 4909 tfi 6688 lebnumlem3 21463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-v 3111 df-dif 3478 df-in 3482 df-ss 3489 df-nul 3785 |
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