Theorem pw1dc1 7011

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 7008	. 2 |- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )
2		elpwi 3625	. . . . 5 |- ( x e. ~P 1o -> x C_ 1o )
3		ss1o0el1o 7010	. . . . 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 840	. . 3 |- ( x e. ~P 1o -> (DECID (/) e. x <-> DECID x = 1o ) )
6	5	ralbiia 2520	. 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 836   = wceq 1373   e. wcel 2176   A.wral 2484   C_ wss 3166   (/)c0 3460   ~Pcpw 3616  EXMIDwem 4238   1oc1o 6495

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-nul 4170

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-exmid 4239  df-suc 4418  df-1o 6502

This theorem is referenced by: (None)