Theorem pw1dc1 7105

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 7102	. 2 |- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )
2		elpwi 3661	. . . . 5 |- ( x e. ~P 1o -> x C_ 1o )
3		ss1o0el1o 7104	. . . . 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 845	. . 3 |- ( x e. ~P 1o -> (DECID (/) e. x <-> DECID x = 1o ) )
6	5	ralbiia 2546	. 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 841   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   (/)c0 3494   ~Pcpw 3652  EXMIDwem 4284   1oc1o 6574

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-exmid 4285  df-suc 4468  df-1o 6581

This theorem is referenced by:  pw1dceq  16605