Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > snon0 | Unicode version |
Description: An ordinal which is a singleton is . (Contributed by Jim Kingdon, 19-Oct-2021.) |
Ref | Expression |
---|---|
snon0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4266 | . . 3 | |
2 | snidg 3400 | . . . . . . 7 | |
3 | 2 | adantr 261 | . . . . . 6 |
4 | ontr1 4126 | . . . . . . 7 | |
5 | 4 | adantl 262 | . . . . . 6 |
6 | 3, 5 | mpan2d 404 | . . . . 5 |
7 | elsni 3393 | . . . . 5 | |
8 | 6, 7 | syl6 29 | . . . 4 |
9 | eleq1 2100 | . . . . 5 | |
10 | 9 | biimpcd 148 | . . . 4 |
11 | 8, 10 | sylcom 25 | . . 3 |
12 | 1, 11 | mtoi 590 | . 2 |
13 | 12 | eq0rdv 3261 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 c0 3224 csn 3375 con0 4100 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |