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Mirrors > Home > NFE Home > Th. List > snel1c | Unicode version |
Description: A singleton is an element of cardinal one. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
snel1c.1 |
Ref | Expression |
---|---|
snel1c | 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2353 | . . 3 | |
2 | snel1c.1 | . . . 4 | |
3 | sneq 3744 | . . . . 5 | |
4 | 3 | eqeq2d 2364 | . . . 4 |
5 | 2, 4 | spcev 2946 | . . 3 |
6 | 1, 5 | ax-mp 5 | . 2 |
7 | el1c 4139 | . 2 1c | |
8 | 6, 7 | mpbir 200 | 1 1c |
Colors of variables: wff setvar class |
Syntax hints: wex 1541 wceq 1642 wcel 1710 cvv 2859 csn 3737 1cc1c 4134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-1c 4136 |
This theorem is referenced by: snel1cg 4141 sikss1c1c 4267 ins2kss 4279 ins3kss 4280 sikexg 4296 ins2kexg 4305 ins3kexg 4306 tfin1c 4499 nnpweq 4523 sfin01 4528 pw1fnval 5851 brpw1fn 5854 df1c3 6140 tc1c 6165 ce0nn 6180 nc0le1 6216 brtcfn 6246 |
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