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Mirrors > Home > ILE Home > Th. List > 3nsssucpw1 | Unicode version |
Description: Negated excluded middle implies that is not a subset of the successor of the power set of . (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.) |
Ref | Expression |
---|---|
3nsssucpw1 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3o 6397 | . . . . . 6 | |
2 | 1 | sseq1i 3173 | . . . . 5 |
3 | 1lt2o 6421 | . . . . . . . . 9 | |
4 | ssnel 4553 | . . . . . . . . 9 | |
5 | 3, 4 | mt2 635 | . . . . . . . 8 |
6 | 2onn 6500 | . . . . . . . . . 10 | |
7 | 6 | elexi 2742 | . . . . . . . . 9 |
8 | 7 | elpw 3572 | . . . . . . . 8 |
9 | 5, 8 | mtbir 666 | . . . . . . 7 |
10 | 9 | a1i 9 | . . . . . 6 |
11 | sucssel 4409 | . . . . . . . . 9 | |
12 | 6, 11 | ax-mp 5 | . . . . . . . 8 |
13 | elsuci 4388 | . . . . . . . 8 | |
14 | 12, 13 | syl 14 | . . . . . . 7 |
15 | 14 | orcomd 724 | . . . . . 6 |
16 | 10, 15 | ecased 1344 | . . . . 5 |
17 | 2, 16 | sylbi 120 | . . . 4 |
18 | 17 | eqcomd 2176 | . . 3 |
19 | exmidpweq 6887 | . . 3 EXMID | |
20 | 18, 19 | sylibr 133 | . 2 EXMID |
21 | 20 | con3i 627 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 703 wceq 1348 wcel 2141 wss 3121 cpw 3566 EXMIDwem 4180 csuc 4350 com 4574 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-int 3832 df-tr 4088 df-exmid 4181 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-1o 6395 df-2o 6396 df-3o 6397 |
This theorem is referenced by: onntri45 7218 |
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