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Mirrors > Home > ILE Home > Th. List > reg2exmid | Unicode version |
Description: If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Ref | Expression |
---|---|
reg2exmid.1 |
Ref | Expression |
---|---|
reg2exmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2081 | . . . 4 | |
2 | 1 | regexmidlemm 4275 | . . 3 |
3 | reg2exmid.1 | . . . 4 | |
4 | pp0ex 3960 | . . . . . 6 | |
5 | 4 | rabex 3922 | . . . . 5 |
6 | eleq2 2142 | . . . . . . 7 | |
7 | 6 | exbidv 1746 | . . . . . 6 |
8 | raleq 2549 | . . . . . . 7 | |
9 | 8 | rexeqbi1dv 2558 | . . . . . 6 |
10 | 7, 9 | imbi12d 232 | . . . . 5 |
11 | 5, 10 | spcv 2691 | . . . 4 |
12 | 3, 11 | ax-mp 7 | . . 3 |
13 | 2, 12 | ax-mp 7 | . 2 |
14 | 1 | reg2exmidlema 4277 | . 2 |
15 | 13, 14 | ax-mp 7 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wo 661 wal 1282 wceq 1284 wex 1421 wcel 1433 wral 2348 wrex 2349 crab 2352 wss 2973 c0 3251 csn 3398 cpr 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 |
This theorem is referenced by: (None) |
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