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Mirrors > Home > ILE Home > Th. List > reg3exmid | Unicode version |
Description: If any inhabited set satisfying df-wetr 4294 for has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.) |
Ref | Expression |
---|---|
reg3exmid.1 |
Ref | Expression |
---|---|
reg3exmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2157 | . . 3 | |
2 | 1 | regexmidlemm 4491 | . 2 |
3 | 1 | reg3exmidlemwe 4538 | . . 3 |
4 | pp0ex 4150 | . . . . 5 | |
5 | 4 | rabex 4108 | . . . 4 |
6 | weeq2 4317 | . . . . . 6 | |
7 | eleq2 2221 | . . . . . . 7 | |
8 | 7 | exbidv 1805 | . . . . . 6 |
9 | 6, 8 | anbi12d 465 | . . . . 5 |
10 | raleq 2652 | . . . . . 6 | |
11 | 10 | rexeqbi1dv 2661 | . . . . 5 |
12 | 9, 11 | imbi12d 233 | . . . 4 |
13 | reg3exmid.1 | . . . 4 | |
14 | 5, 12, 13 | vtocl 2766 | . . 3 |
15 | 3, 14 | mpan 421 | . 2 |
16 | 1 | reg2exmidlema 4493 | . 2 |
17 | 2, 15, 16 | mp2b 8 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 wceq 1335 wex 1472 wcel 2128 wral 2435 wrex 2436 crab 2439 wss 3102 c0 3394 csn 3560 cpr 3561 cep 4247 wwe 4290 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-eprel 4249 df-frfor 4291 df-frind 4292 df-wetr 4294 |
This theorem is referenced by: (None) |
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