Theorem pw1dceq 16765

Description: The powerset of 1o having decidable equality is equivalent to excluded middle. (Contributed by Jim Kingdon, 12-Feb-2026.)

Assertion

Ref	Expression
pw1dceq	|- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1oDECID x = y )

Distinct variable group:   x, y

Proof of Theorem pw1dceq

Step	Hyp	Ref	Expression
1		exmidexmid 4308	. . . 4 |- (EXMID -> DECID x = y )
2	1	ralrimivw 2616	. . 3 |- (EXMID -> A. y e. ~P 1oDECID x = y )
3	2	ralrimivw 2616	. 2 |- (EXMID -> A. x e. ~P 1o A. y e. ~P 1oDECID x = y )
4		1oex 6654	. . . . . 6 |- 1o e. _V
5	4	pwid 3686	. . . . 5 |- 1o e. ~P 1o
6		eqeq2 2242	. . . . . . 7 |- ( y = 1o -> ( x = y <-> x = 1o ) )
7	6	dcbid 846	. . . . . 6 |- ( y = 1o -> (DECID x = y <-> DECID x = 1o ) )
8	7	rspcv 2916	. . . . 5 |- ( 1o e. ~P 1o -> ( A. y e. ~P 1oDECID x = y -> DECID x = 1o ) )
9	5, 8	ax-mp 5	. . . 4 |- ( A. y e. ~P 1oDECID x = y -> DECID x = 1o )
10	9	ralimi 2605	. . 3 |- ( A. x e. ~P 1o A. y e. ~P 1oDECID x = y -> A. x e. ~P 1oDECID x = 1o )
11		pw1dc1 7173	. . 3 |- (EXMID <-> A. x e. ~P 1oDECID x = 1o )
12	10, 11	sylibr 134	. 2 |- ( A. x e. ~P 1o A. y e. ~P 1oDECID x = y -> EXMID )
13	3, 12	impbii 126	1 |- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1oDECID x = y )

Syntax hints:   -> wi 4   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2203   A.wral 2520   ~Pcpw 3668  EXMIDwem 4306   1oc1o 6639

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-uni 3914  df-tr 4208  df-exmid 4307  df-iord 4486  df-on 4488  df-suc 4491  df-1o 6646

This theorem is referenced by: (None)