Theorem pw1dceq 16665

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 4288	. . . 4 |- (EXMID -> DECID x = y )
2	1	ralrimivw 2605	. . 3 |- (EXMID -> A. y e. ~P 1oDECID x = y )
3	2	ralrimivw 2605	. 2 |- (EXMID -> A. x e. ~P 1o A. y e. ~P 1oDECID x = y )
4		1oex 6595	. . . . . 6 |- 1o e. _V
5	4	pwid 3668	. . . . 5 |- 1o e. ~P 1o
6		eqeq2 2240	. . . . . . 7 |- ( y = 1o -> ( x = y <-> x = 1o ) )
7	6	dcbid 845	. . . . . 6 |- ( y = 1o -> (DECID x = y <-> DECID x = 1o ) )
8	7	rspcv 2905	. . . . 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 2594	. . 3 |- ( A. x e. ~P 1o A. y e. ~P 1oDECID x = y -> A. x e. ~P 1oDECID x = 1o )
11		pw1dc1 7111	. . 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 841   = wceq 1397   e. wcel 2201   A.wral 2509   ~Pcpw 3653  EXMIDwem 4286   1oc1o 6580

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-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-uni 3895  df-tr 4189  df-exmid 4287  df-iord 4465  df-on 4467  df-suc 4470  df-1o 6587

This theorem is referenced by: (None)