Theorem pw1dc1 7211

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 7208	. 2 |- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )
2		elpwi 3694	. . . . 5 |- ( x e. ~P 1o -> x C_ 1o )
3		ss1o0el1o 7210	. . . . 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 850	. . 3 |- ( x e. ~P 1o -> (DECID (/) e. x <-> DECID x = 1o ) )
6	5	ralbiia 2564	. 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 846   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   (/)c0 3520   ~Pcpw 3685  EXMIDwem 4326   1oc1o 6670

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-exmid 4327  df-suc 4511  df-1o 6677

This theorem is referenced by:  pw1dceq  16948