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Mirrors > Home > ILE Home > Th. List > reg3exmid | Unicode version |
Description: If any inhabited set satisfying df-wetr 4319 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 2170 | . . 3 | |
2 | 1 | regexmidlemm 4516 | . 2 |
3 | 1 | reg3exmidlemwe 4563 | . . 3 |
4 | pp0ex 4175 | . . . . 5 | |
5 | 4 | rabex 4133 | . . . 4 |
6 | weeq2 4342 | . . . . . 6 | |
7 | eleq2 2234 | . . . . . . 7 | |
8 | 7 | exbidv 1818 | . . . . . 6 |
9 | 6, 8 | anbi12d 470 | . . . . 5 |
10 | raleq 2665 | . . . . . 6 | |
11 | 10 | rexeqbi1dv 2674 | . . . . 5 |
12 | 9, 11 | imbi12d 233 | . . . 4 |
13 | reg3exmid.1 | . . . 4 | |
14 | 5, 12, 13 | vtocl 2784 | . . 3 |
15 | 3, 14 | mpan 422 | . 2 |
16 | 1 | reg2exmidlema 4518 | . 2 |
17 | 2, 15, 16 | mp2b 8 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 wceq 1348 wex 1485 wcel 2141 wral 2448 wrex 2449 crab 2452 wss 3121 c0 3414 csn 3583 cpr 3584 cep 4272 wwe 4315 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-eprel 4274 df-frfor 4316 df-frind 4317 df-wetr 4319 |
This theorem is referenced by: (None) |
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