Theorem pw1dc1 6859

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.)

Assertion

Ref	Expression
pw1dc1	|- (EXMID <-> A. x e. ~P 1oDECID x = 1o )

Proof of Theorem pw1dc1

Step	Hyp	Ref	Expression
1		pw1dc0el 6857	. 2 |- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )
2		elpwi 3552	. . . . 5 |- ( x e. ~P 1o -> x C_ 1o )
3		ss1o0el1o 6858	. . . . 5 |- ( x C_ 1o -> ( (/) e. x <-> x = 1o ) )
4	2, 3	syl 14	. . . 4 |- ( x e. ~P 1o -> ( (/) e. x <-> x = 1o ) )
5	4	dcbid 824	. . 3 |- ( x e. ~P 1o -> (DECID (/) e. x <-> DECID x = 1o ) )
6	5	ralbiia 2471	. 2 |- ( A. x e. ~P 1oDECID (/) e. x <-> A. x e. ~P 1oDECID x = 1o )
7	1, 6	bitri 183	1 |- (EXMID <-> A. x e. ~P 1oDECID x = 1o )

Syntax hints:   <-> wb 104  DECID wdc 820   = wceq 1335   e. wcel 2128   A.wral 2435   C_ wss 3102   (/)c0 3394   ~Pcpw 3543  EXMIDwem 4156   1oc1o 6357

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-ext 2139  ax-nul 4091

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  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-exmid 4157  df-suc 4332  df-1o 6364

This theorem is referenced by: (None)