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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 2040 | . . . 4 | |
2 | 1 | regexmidlemm 4257 | . . 3 |
3 | reg2exmid.1 | . . . 4 | |
4 | pp0ex 3940 | . . . . . 6 | |
5 | 4 | rabex 3901 | . . . . 5 |
6 | eleq2 2101 | . . . . . . 7 | |
7 | 6 | exbidv 1706 | . . . . . 6 |
8 | raleq 2505 | . . . . . . 7 | |
9 | 8 | rexeqbi1dv 2514 | . . . . . 6 |
10 | 7, 9 | imbi12d 223 | . . . . 5 |
11 | 5, 10 | spcv 2646 | . . . 4 |
12 | 3, 11 | ax-mp 7 | . . 3 |
13 | 2, 12 | ax-mp 7 | . 2 |
14 | 1 | reg2exmidlema 4259 | . 2 |
15 | 13, 14 | ax-mp 7 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 wal 1241 wceq 1243 wex 1381 wcel 1393 wral 2306 wrex 2307 crab 2310 wss 2917 c0 3224 csn 3375 cpr 3376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 |
This theorem is referenced by: (None) |
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