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Mirrors > Home > ILE Home > Th. List > pw1dc1 | Unicode version |
Description: If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.) |
Ref | Expression |
---|---|
pw1dc1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw1dc0el 6908 |
. 2
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2 | elpwi 3584 |
. . . . 5
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3 | ss1o0el1o 6909 |
. . . . 5
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4 | 2, 3 | syl 14 |
. . . 4
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5 | 4 | dcbid 838 |
. . 3
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6 | 5 | ralbiia 2491 |
. 2
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7 | 1, 6 | bitri 184 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-nul 4128 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-exmid 4194 df-suc 4370 df-1o 6414 |
This theorem is referenced by: (None) |
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