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Mirrors > Home > ILE Home > Th. List > exmidpw | Unicode version |
Description: Excluded middle is equivalent to the power set of having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.) |
Ref | Expression |
---|---|
exmidpw | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 6408 | . . . . 5 | |
2 | p0ex 4174 | . . . . 5 | |
3 | 1, 2 | eqeltri 2243 | . . . 4 |
4 | 3 | pwex 4169 | . . 3 |
5 | exmid01 4184 | . . . . . . . . 9 EXMID | |
6 | 5 | biimpi 119 | . . . . . . . 8 EXMID |
7 | 6 | 19.21bi 1551 | . . . . . . 7 EXMID |
8 | 1 | pweqi 3570 | . . . . . . . . 9 |
9 | 8 | eleq2i 2237 | . . . . . . . 8 |
10 | velpw 3573 | . . . . . . . 8 | |
11 | 9, 10 | bitri 183 | . . . . . . 7 |
12 | vex 2733 | . . . . . . . 8 | |
13 | 12 | elpr 3604 | . . . . . . 7 |
14 | 7, 11, 13 | 3imtr4g 204 | . . . . . 6 EXMID |
15 | 14 | ssrdv 3153 | . . . . 5 EXMID |
16 | pwpw0ss 3791 | . . . . . . 7 | |
17 | 16, 8 | sseqtrri 3182 | . . . . . 6 |
18 | 17 | a1i 9 | . . . . 5 EXMID |
19 | 15, 18 | eqssd 3164 | . . . 4 EXMID |
20 | df2o2 6410 | . . . 4 | |
21 | 19, 20 | eqtr4di 2221 | . . 3 EXMID |
22 | eqeng 6744 | . . 3 | |
23 | 4, 21, 22 | mpsyl 65 | . 2 EXMID |
24 | 0nep0 4151 | . . . . . . . 8 | |
25 | 0ex 4116 | . . . . . . . . . . 11 | |
26 | 25, 2 | prss 3736 | . . . . . . . . . 10 |
27 | 17, 26 | mpbir 145 | . . . . . . . . 9 |
28 | en2eqpr 6885 | . . . . . . . . . 10 | |
29 | 28 | 3expb 1199 | . . . . . . . . 9 |
30 | 27, 29 | mpan2 423 | . . . . . . . 8 |
31 | 24, 30 | mpi 15 | . . . . . . 7 |
32 | 31 | eleq2d 2240 | . . . . . 6 |
33 | 32, 11, 13 | 3bitr3g 221 | . . . . 5 |
34 | 33 | biimpd 143 | . . . 4 |
35 | 34 | alrimiv 1867 | . . 3 |
36 | 35, 5 | sylibr 133 | . 2 EXMID |
37 | 23, 36 | 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 wne 2340 cvv 2730 wss 3121 c0 3414 cpw 3566 csn 3583 cpr 3584 class class class wbr 3989 EXMIDwem 4180 c1o 6388 c2o 6389 cen 6716 |
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 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 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-sbc 2956 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-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-exmid 4181 df-id 4278 df-suc 4356 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-1o 6395 df-2o 6396 df-en 6719 |
This theorem is referenced by: pwf1oexmid 14032 |
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