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Mirrors > Home > ILE Home > Th. List > exmidpweq | Unicode version |
Description: Excluded middle is equivalent to the power set of being . (Contributed by Jim Kingdon, 28-Jul-2024.) |
Ref | Expression |
---|---|
exmidpweq | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid01 4160 | . . . . . . . 8 EXMID | |
2 | 1 | biimpi 119 | . . . . . . 7 EXMID |
3 | 2 | 19.21bi 1538 | . . . . . 6 EXMID |
4 | df1o2 6377 | . . . . . . . . 9 | |
5 | 4 | pweqi 3547 | . . . . . . . 8 |
6 | 5 | eleq2i 2224 | . . . . . . 7 |
7 | velpw 3550 | . . . . . . 7 | |
8 | 6, 7 | bitri 183 | . . . . . 6 |
9 | vex 2715 | . . . . . . 7 | |
10 | 9 | elpr 3581 | . . . . . 6 |
11 | 3, 8, 10 | 3imtr4g 204 | . . . . 5 EXMID |
12 | 11 | ssrdv 3134 | . . . 4 EXMID |
13 | pwpw0ss 3768 | . . . . . 6 | |
14 | 13, 5 | sseqtrri 3163 | . . . . 5 |
15 | 14 | a1i 9 | . . . 4 EXMID |
16 | 12, 15 | eqssd 3145 | . . 3 EXMID |
17 | df2o2 6379 | . . 3 | |
18 | 16, 17 | eqtr4di 2208 | . 2 EXMID |
19 | simpr 109 | . . . . . . . . 9 | |
20 | 19, 7 | sylibr 133 | . . . . . . . 8 |
21 | 20, 5 | eleqtrrdi 2251 | . . . . . . 7 |
22 | simpl 108 | . . . . . . . 8 | |
23 | 22, 17 | eqtrdi 2206 | . . . . . . 7 |
24 | 21, 23 | eleqtrd 2236 | . . . . . 6 |
25 | 24, 10 | sylib 121 | . . . . 5 |
26 | 25 | ex 114 | . . . 4 |
27 | 26 | alrimiv 1854 | . . 3 |
28 | 27, 1 | sylibr 133 | . 2 EXMID |
29 | 18, 28 | impbii 125 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wal 1333 wceq 1335 wcel 2128 wss 3102 c0 3394 cpw 3543 csn 3560 cpr 3561 EXMIDwem 4156 c1o 6357 c2o 6358 |
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-nul 4091 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 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-exmid 4157 df-suc 4332 df-1o 6364 df-2o 6365 |
This theorem is referenced by: pw1fin 6856 pw1nel3 7167 3nsssucpw1 7172 onntri35 7173 |
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