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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 7164 | . . . . 5 | |
2 | pw1ne1 7165 | . . . . 5 | |
3 | 1, 2 | nelpri 3584 | . . . 4 |
4 | 3 | a1i 9 | . . 3 EXMID |
5 | df2o3 6378 | . . . 4 | |
6 | 5 | eleq2i 2224 | . . 3 |
7 | 4, 6 | sylnibr 667 | . 2 EXMID |
8 | exmidpweq 6855 | . . . 4 EXMID | |
9 | 8 | notbii 658 | . . 3 EXMID |
10 | 1oex 6372 | . . . . . 6 | |
11 | 10 | pwex 4145 | . . . . 5 |
12 | 11 | elsn 3576 | . . . 4 |
13 | 12 | notbii 658 | . . 3 |
14 | 9, 13 | sylbb2 137 | . 2 EXMID |
15 | df-3o 6366 | . . . . . . 7 | |
16 | df-suc 4332 | . . . . . . 7 | |
17 | 15, 16 | eqtri 2178 | . . . . . 6 |
18 | 17 | eleq2i 2224 | . . . . 5 |
19 | elun 3248 | . . . . 5 | |
20 | 18, 19 | bitri 183 | . . . 4 |
21 | 20 | notbii 658 | . . 3 |
22 | ioran 742 | . . 3 | |
23 | 21, 22 | bitri 183 | . 2 |
24 | 7, 14, 23 | sylanbrc 414 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 wceq 1335 wcel 2128 cun 3100 c0 3394 cpw 3543 csn 3560 cpr 3561 EXMIDwem 4156 csuc 4326 c1o 6357 c2o 6358 c3o 6359 |
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-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 |
This theorem depends on definitions: df-bi 116 df-dc 821 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-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3774 df-tr 4064 df-exmid 4157 df-iord 4327 df-on 4329 df-suc 4332 df-1o 6364 df-2o 6365 df-3o 6366 |
This theorem is referenced by: sucpw1ne3 7168 sucpw1nss3 7171 |
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