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Mirrors > Home > MPE Home > Th. List > 2mo | Structured version Visualization version Unicode version |
Description: Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.) |
Ref | Expression |
---|---|
2mo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2mo2 2550 | . . . 4 | |
2 | nfmo1 2481 | . . . . . . 7 | |
3 | nfe1 2027 | . . . . . . . 8 | |
4 | 3 | nfmo 2487 | . . . . . . 7 |
5 | 2, 4 | nfan 1828 | . . . . . 6 |
6 | nfe1 2027 | . . . . . . . . 9 | |
7 | 6 | nfmo 2487 | . . . . . . . 8 |
8 | nfmo1 2481 | . . . . . . . 8 | |
9 | 7, 8 | nfan 1828 | . . . . . . 7 |
10 | 19.8a 2052 | . . . . . . . . 9 | |
11 | spsbe 1884 | . . . . . . . . . 10 | |
12 | 11 | sbimi 1886 | . . . . . . . . 9 |
13 | nfv 1843 | . . . . . . . . . . . 12 | |
14 | 13 | mo3 2507 | . . . . . . . . . . 11 |
15 | 14 | biimpi 206 | . . . . . . . . . 10 |
16 | 15 | 19.21bbi 2060 | . . . . . . . . 9 |
17 | 10, 12, 16 | syl2ani 688 | . . . . . . . 8 |
18 | 19.8a 2052 | . . . . . . . . 9 | |
19 | sbcom2 2445 | . . . . . . . . . 10 | |
20 | spsbe 1884 | . . . . . . . . . . 11 | |
21 | 20 | sbimi 1886 | . . . . . . . . . 10 |
22 | 19, 21 | sylbi 207 | . . . . . . . . 9 |
23 | nfv 1843 | . . . . . . . . . . . 12 | |
24 | 23 | mo3 2507 | . . . . . . . . . . 11 |
25 | 24 | biimpi 206 | . . . . . . . . . 10 |
26 | 25 | 19.21bbi 2060 | . . . . . . . . 9 |
27 | 18, 22, 26 | syl2ani 688 | . . . . . . . 8 |
28 | 17, 27 | anim12ii 594 | . . . . . . 7 |
29 | 9, 28 | alrimi 2082 | . . . . . 6 |
30 | 5, 29 | alrimi 2082 | . . . . 5 |
31 | 30 | alrimivv 1856 | . . . 4 |
32 | 1, 31 | sylbir 225 | . . 3 |
33 | nfs1v 2437 | . . . . . . . 8 | |
34 | nfs1v 2437 | . . . . . . . . . 10 | |
35 | 34 | nfsb 2440 | . . . . . . . . 9 |
36 | pm3.21 464 | . . . . . . . . . 10 | |
37 | 36 | imim1d 82 | . . . . . . . . 9 |
38 | 35, 37 | alimd 2081 | . . . . . . . 8 |
39 | 33, 38 | alimd 2081 | . . . . . . 7 |
40 | 39 | com12 32 | . . . . . 6 |
41 | 40 | aleximi 1759 | . . . . 5 |
42 | 41 | aleximi 1759 | . . . 4 |
43 | 2nexaln 1757 | . . . . . 6 | |
44 | 2sb8e 2467 | . . . . . 6 | |
45 | 43, 44 | xchnxbi 322 | . . . . 5 |
46 | pm2.21 120 | . . . . . . . . 9 | |
47 | 46 | 2alimi 1740 | . . . . . . . 8 |
48 | 47 | 2eximi 1763 | . . . . . . 7 |
49 | 48 | 19.23bi 2061 | . . . . . 6 |
50 | 49 | 19.23bi 2061 | . . . . 5 |
51 | 45, 50 | sylbi 207 | . . . 4 |
52 | 42, 51 | pm2.61d1 171 | . . 3 |
53 | 32, 52 | impbii 199 | . 2 |
54 | alrot4 2039 | . 2 | |
55 | 53, 54 | bitri 264 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wal 1481 wex 1704 wsb 1880 wmo 2471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 |
This theorem is referenced by: 2mos 2552 |
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