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Mirrors > Home > NFE Home > Th. List > 0nelsuc | Unicode version |
Description: The empty class is not a member of a successor. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
0nelsuc | 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | el1c 4139 | . . . . . . . . 9 1c | |
2 | vex 2862 | . . . . . . . . . . . . 13 | |
3 | 2 | snid 3760 | . . . . . . . . . . . 12 |
4 | n0i 3555 | . . . . . . . . . . . 12 | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . . . 11 |
6 | eqeq1 2359 | . . . . . . . . . . 11 | |
7 | 5, 6 | mtbiri 294 | . . . . . . . . . 10 |
8 | 7 | exlimiv 1634 | . . . . . . . . 9 |
9 | 1, 8 | sylbi 187 | . . . . . . . 8 1c |
10 | simpr 447 | . . . . . . . 8 | |
11 | 9, 10 | nsyl 113 | . . . . . . 7 1c |
12 | un00 3586 | . . . . . . . . 9 | |
13 | eqcom 2355 | . . . . . . . . 9 | |
14 | 12, 13 | bitri 240 | . . . . . . . 8 |
15 | 14 | notbii 287 | . . . . . . 7 |
16 | 11, 15 | sylib 188 | . . . . . 6 1c |
17 | simpr 447 | . . . . . 6 | |
18 | 16, 17 | nsyl 113 | . . . . 5 1c |
19 | 18 | nrex 2716 | . . . 4 1c |
20 | 19 | a1i 10 | . . 3 1c |
21 | 20 | nrex 2716 | . 2 1c |
22 | eladdc 4398 | . 2 1c 1c | |
23 | 21, 22 | mtbir 290 | 1 1c |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wa 358 wex 1541 wceq 1642 wcel 1710 wrex 2615 cun 3207 cin 3208 c0 3550 csn 3737 1cc1c 4134 cplc 4375 |
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-ral 2619 df-rex 2620 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 df-addc 4378 |
This theorem is referenced by: 0cnsuc 4401 nndisjeq 4429 sfinltfin 4535 |
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