Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6362 | . . . . . 6 | |
2 | 1 | sseq1i 3154 | . . . . 5 |
3 | 1lt2o 6386 | . . . . . . . . 9 | |
4 | ssnel 4527 | . . . . . . . . 9 | |
5 | 3, 4 | mt2 630 | . . . . . . . 8 |
6 | 2onn 6465 | . . . . . . . . . 10 | |
7 | 6 | elexi 2724 | . . . . . . . . 9 |
8 | 7 | elpw 3549 | . . . . . . . 8 |
9 | 5, 8 | mtbir 661 | . . . . . . 7 |
10 | 9 | a1i 9 | . . . . . 6 |
11 | sucssel 4384 | . . . . . . . . 9 | |
12 | 6, 11 | ax-mp 5 | . . . . . . . 8 |
13 | elsuci 4363 | . . . . . . . 8 | |
14 | 12, 13 | syl 14 | . . . . . . 7 |
15 | 14 | orcomd 719 | . . . . . 6 |
16 | 10, 15 | ecased 1331 | . . . . 5 |
17 | 2, 16 | sylbi 120 | . . . 4 |
18 | 17 | eqcomd 2163 | . . 3 |
19 | exmidpweq 6851 | . . 3 EXMID | |
20 | 18, 19 | sylibr 133 | . 2 EXMID |
21 | 20 | con3i 622 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 698 wceq 1335 wcel 2128 wss 3102 cpw 3543 EXMIDwem 4155 csuc 4325 com 4548 c1o 6353 c2o 6354 c3o 6355 |
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 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 |
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 3773 df-int 3808 df-tr 4063 df-exmid 4156 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-1o 6360 df-2o 6361 df-3o 6362 |
This theorem is referenced by: onntri45 7171 |
Copyright terms: Public domain | W3C validator |