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Mirrors > Home > ILE Home > Th. List > onunsnss | Unicode version |
Description: Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.) |
Ref | Expression |
---|---|
onunsnss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4266 | . . . . 5 | |
2 | elsni 3393 | . . . . . . . 8 | |
3 | 2 | adantl 262 | . . . . . . 7 |
4 | simplr 482 | . . . . . . 7 | |
5 | 3, 4 | eqeltrrd 2115 | . . . . . 6 |
6 | 5 | ex 108 | . . . . 5 |
7 | 1, 6 | mtoi 590 | . . . 4 |
8 | snidg 3400 | . . . . . . . . 9 | |
9 | elun2 3111 | . . . . . . . . 9 | |
10 | 8, 9 | syl 14 | . . . . . . . 8 |
11 | 10 | adantr 261 | . . . . . . 7 |
12 | ontr1 4126 | . . . . . . . 8 | |
13 | 12 | adantl 262 | . . . . . . 7 |
14 | 11, 13 | mpan2d 404 | . . . . . 6 |
15 | 14 | imp 115 | . . . . 5 |
16 | elun 3084 | . . . . 5 | |
17 | 15, 16 | sylib 127 | . . . 4 |
18 | 7, 17 | ecased 1239 | . . 3 |
19 | 18 | ex 108 | . 2 |
20 | 19 | ssrdv 2951 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 wceq 1243 wcel 1393 cun 2915 wss 2917 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-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 |
This theorem is referenced by: (None) |
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