Theorem pw1dc1 6984

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 6981	. 2 |- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )
2		elpwi 3615	. . . . 5 |- ( x e. ~P 1o -> x C_ 1o )
3		ss1o0el1o 6983	. . . . 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 839	. . 3 |- ( x e. ~P 1o -> (DECID (/) e. x <-> DECID x = 1o ) )
6	5	ralbiia 2511	. 2 |- ( A. x e. ~P 1oDECID (/) e. x <-> A. x e. ~P 1oDECID x = 1o )
7	1, 6	bitri 184	1 |- (EXMID <-> A. x e. ~P 1oDECID x = 1o )

Syntax hints:   <-> wb 105  DECID wdc 835   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   (/)c0 3451   ~Pcpw 3606  EXMIDwem 4228   1oc1o 6476

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4160

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-exmid 4229  df-suc 4407  df-1o 6483

This theorem is referenced by: (None)