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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 4184 | . . . . . . . 8 EXMID | |
2 | 1 | biimpi 119 | . . . . . . 7 EXMID |
3 | 2 | 19.21bi 1551 | . . . . . 6 EXMID |
4 | df1o2 6408 | . . . . . . . . 9 | |
5 | 4 | pweqi 3570 | . . . . . . . 8 |
6 | 5 | eleq2i 2237 | . . . . . . 7 |
7 | velpw 3573 | . . . . . . 7 | |
8 | 6, 7 | bitri 183 | . . . . . 6 |
9 | vex 2733 | . . . . . . 7 | |
10 | 9 | elpr 3604 | . . . . . 6 |
11 | 3, 8, 10 | 3imtr4g 204 | . . . . 5 EXMID |
12 | 11 | ssrdv 3153 | . . . 4 EXMID |
13 | pwpw0ss 3791 | . . . . . 6 | |
14 | 13, 5 | sseqtrri 3182 | . . . . 5 |
15 | 14 | a1i 9 | . . . 4 EXMID |
16 | 12, 15 | eqssd 3164 | . . 3 EXMID |
17 | df2o2 6410 | . . 3 | |
18 | 16, 17 | eqtr4di 2221 | . 2 EXMID |
19 | simpr 109 | . . . . . . . . 9 | |
20 | 19, 7 | sylibr 133 | . . . . . . . 8 |
21 | 20, 5 | eleqtrrdi 2264 | . . . . . . 7 |
22 | simpl 108 | . . . . . . . 8 | |
23 | 22, 17 | eqtrdi 2219 | . . . . . . 7 |
24 | 21, 23 | eleqtrd 2249 | . . . . . 6 |
25 | 24, 10 | sylib 121 | . . . . 5 |
26 | 25 | ex 114 | . . . 4 |
27 | 26 | alrimiv 1867 | . . 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 703 wal 1346 wceq 1348 wcel 2141 wss 3121 c0 3414 cpw 3566 csn 3583 cpr 3584 EXMIDwem 4180 c1o 6388 c2o 6389 |
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-ext 2152 ax-nul 4115 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 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-exmid 4181 df-suc 4356 df-1o 6395 df-2o 6396 |
This theorem is referenced by: pw1fin 6888 pw1nel3 7208 3nsssucpw1 7213 onntri35 7214 |
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