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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 2137 | . . . 4 | |
2 | 1 | regexmidlemm 4442 | . . 3 |
3 | reg2exmid.1 | . . . 4 | |
4 | pp0ex 4108 | . . . . . 6 | |
5 | 4 | rabex 4067 | . . . . 5 |
6 | eleq2 2201 | . . . . . . 7 | |
7 | 6 | exbidv 1797 | . . . . . 6 |
8 | raleq 2624 | . . . . . . 7 | |
9 | 8 | rexeqbi1dv 2633 | . . . . . 6 |
10 | 7, 9 | imbi12d 233 | . . . . 5 |
11 | 5, 10 | spcv 2774 | . . . 4 |
12 | 3, 11 | ax-mp 5 | . . 3 |
13 | 2, 12 | ax-mp 5 | . 2 |
14 | 1 | reg2exmidlema 4444 | . 2 |
15 | 13, 14 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 wal 1329 wceq 1331 wex 1468 wcel 1480 wral 2414 wrex 2415 crab 2418 wss 3066 c0 3358 csn 3522 cpr 3523 |
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-rex 2420 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 df-pr 3529 |
This theorem is referenced by: (None) |
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