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Mirrors > Home > ILE Home > Th. List > undifexmid | Unicode version |
Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3438 and undifdcss 6804 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.) |
Ref | Expression |
---|---|
undifexmid.1 |
Ref | Expression |
---|---|
undifexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4050 | . . . . 5 | |
2 | 1 | snid 3551 | . . . 4 |
3 | ssrab2 3177 | . . . . 5 | |
4 | p0ex 4107 | . . . . . . 7 | |
5 | 4 | rabex 4067 | . . . . . 6 |
6 | sseq12 3117 | . . . . . . 7 | |
7 | simpl 108 | . . . . . . . . 9 | |
8 | simpr 109 | . . . . . . . . . 10 | |
9 | 8, 7 | difeq12d 3190 | . . . . . . . . 9 |
10 | 7, 9 | uneq12d 3226 | . . . . . . . 8 |
11 | 10, 8 | eqeq12d 2152 | . . . . . . 7 |
12 | 6, 11 | bibi12d 234 | . . . . . 6 |
13 | undifexmid.1 | . . . . . 6 | |
14 | 5, 4, 12, 13 | vtocl2 2736 | . . . . 5 |
15 | 3, 14 | mpbi 144 | . . . 4 |
16 | 2, 15 | eleqtrri 2213 | . . 3 |
17 | elun 3212 | . . 3 | |
18 | 16, 17 | mpbi 144 | . 2 |
19 | biidd 171 | . . . . . 6 | |
20 | 19 | elrab3 2836 | . . . . 5 |
21 | 2, 20 | ax-mp 5 | . . . 4 |
22 | 21 | biimpi 119 | . . 3 |
23 | eldifn 3194 | . . . 4 | |
24 | 23, 21 | sylnib 665 | . . 3 |
25 | 22, 24 | orim12i 748 | . 2 |
26 | 18, 25 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 crab 2418 cdif 3063 cun 3064 wss 3066 c0 3358 csn 3522 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 |
This theorem is referenced by: (None) |
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