Description: There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lhpex1.l | |
|
lhpex1.a | |
||
lhpex1.h | |
||
Assertion | lhpexle3 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpex1.l | |
|
2 | lhpex1.a | |
|
3 | lhpex1.h | |
|
4 | 1 2 3 | lhpexle2 | |
5 | 3anass | |
|
6 | 5 | rexbii | |
7 | 4 6 | sylib | |
8 | 1 2 3 | lhpexle2 | |
9 | 8 | adantr | |
10 | 3anass | |
|
11 | 10 | rexbii | |
12 | 9 11 | sylib | |
13 | 1 2 3 | lhpexle2 | |
14 | 3anass | |
|
15 | 14 | rexbii | |
16 | 13 15 | sylib | |
17 | 16 | 3ad2ant1 | |
18 | simpl1 | |
|
19 | simpl3l | |
|
20 | simpl2l | |
|
21 | simprl | |
|
22 | simpl3r | |
|
23 | simpl2r | |
|
24 | simprr | |
|
25 | 1 2 3 | lhpexle3lem | |
26 | 18 19 20 21 22 23 24 25 | syl133anc | |
27 | df-3an | |
|
28 | 27 | anbi2i | |
29 | 3anass | |
|
30 | 28 29 | bitr4i | |
31 | 30 | rexbii | |
32 | 26 31 | sylib | |
33 | 17 32 | lhpexle1lem | |
34 | an31 | |
|
35 | 34 | anbi2i | |
36 | 3anass | |
|
37 | 35 29 36 | 3bitr4i | |
38 | 37 | rexbii | |
39 | 33 38 | sylib | |
40 | 39 | 3expa | |
41 | 12 40 | lhpexle1lem | |
42 | an32 | |
|
43 | 42 | anbi2i | |
44 | 3anass | |
|
45 | 43 36 44 | 3bitr4i | |
46 | 45 | rexbii | |
47 | 41 46 | sylib | |
48 | 7 47 | lhpexle1lem | |
49 | df-3an | |
|
50 | 49 | anbi2i | |
51 | 44 50 | bitr4i | |
52 | 51 | rexbii | |
53 | 48 52 | sylib | |