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Mirrors > Home > ILE Home > Th. List > pw1nel3 | Unicode version |
Description: Negated excluded middle implies that the power set of is not an element of . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
Ref | Expression |
---|---|
pw1nel3 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw1ne0 7205 | . . . . 5 | |
2 | pw1ne1 7206 | . . . . 5 | |
3 | 1, 2 | nelpri 3607 | . . . 4 |
4 | 3 | a1i 9 | . . 3 EXMID |
5 | df2o3 6409 | . . . 4 | |
6 | 5 | eleq2i 2237 | . . 3 |
7 | 4, 6 | sylnibr 672 | . 2 EXMID |
8 | exmidpweq 6887 | . . . 4 EXMID | |
9 | 8 | notbii 663 | . . 3 EXMID |
10 | 1oex 6403 | . . . . . 6 | |
11 | 10 | pwex 4169 | . . . . 5 |
12 | 11 | elsn 3599 | . . . 4 |
13 | 12 | notbii 663 | . . 3 |
14 | 9, 13 | sylbb2 137 | . 2 EXMID |
15 | df-3o 6397 | . . . . . . 7 | |
16 | df-suc 4356 | . . . . . . 7 | |
17 | 15, 16 | eqtri 2191 | . . . . . 6 |
18 | 17 | eleq2i 2237 | . . . . 5 |
19 | elun 3268 | . . . . 5 | |
20 | 18, 19 | bitri 183 | . . . 4 |
21 | 20 | notbii 663 | . . 3 |
22 | ioran 747 | . . 3 | |
23 | 21, 22 | bitri 183 | . 2 |
24 | 7, 14, 23 | sylanbrc 415 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 wceq 1348 wcel 2141 cun 3119 c0 3414 cpw 3566 csn 3583 cpr 3584 EXMIDwem 4180 csuc 4350 c1o 6388 c2o 6389 c3o 6390 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-tr 4088 df-exmid 4181 df-iord 4351 df-on 4353 df-suc 4356 df-1o 6395 df-2o 6396 df-3o 6397 |
This theorem is referenced by: sucpw1ne3 7209 sucpw1nss3 7212 |
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