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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 4501 | . . . . 5 | |
2 | elsni 3578 | . . . . . . . 8 | |
3 | 2 | adantl 275 | . . . . . . 7 |
4 | simplr 520 | . . . . . . 7 | |
5 | 3, 4 | eqeltrrd 2235 | . . . . . 6 |
6 | 5 | ex 114 | . . . . 5 |
7 | 1, 6 | mtoi 654 | . . . 4 |
8 | snidg 3589 | . . . . . . . . 9 | |
9 | elun2 3275 | . . . . . . . . 9 | |
10 | 8, 9 | syl 14 | . . . . . . . 8 |
11 | 10 | adantr 274 | . . . . . . 7 |
12 | ontr1 4350 | . . . . . . . 8 | |
13 | 12 | adantl 275 | . . . . . . 7 |
14 | 11, 13 | mpan2d 425 | . . . . . 6 |
15 | 14 | imp 123 | . . . . 5 |
16 | elun 3248 | . . . . 5 | |
17 | 15, 16 | sylib 121 | . . . 4 |
18 | 7, 17 | ecased 1331 | . . 3 |
19 | 18 | ex 114 | . 2 |
20 | 19 | ssrdv 3134 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wcel 2128 cun 3100 wss 3102 csn 3560 con0 4324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-setind 4497 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-uni 3774 df-tr 4064 df-iord 4327 df-on 4329 |
This theorem is referenced by: (None) |
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