Theorem pw1dceq 17017

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 4331	. . . 4 |- (EXMID -> DECID x = y )
2	1	ralrimivw 2624	. . 3 |- (EXMID -> A. y e. ~P 1oDECID x = y )
3	2	ralrimivw 2624	. 2 |- (EXMID -> A. x e. ~P 1o A. y e. ~P 1oDECID x = y )
4		1oex 6689	. . . . . 6 |- 1o e. _V
5	4	pwid 3706	. . . . 5 |- 1o e. ~P 1o
6		eqeq2 2248	. . . . . . 7 |- ( y = 1o -> ( x = y <-> x = 1o ) )
7	6	dcbid 850	. . . . . 6 |- ( y = 1o -> (DECID x = y <-> DECID x = 1o ) )
8	7	rspcv 2925	. . . . 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 2613	. . 3 |- ( A. x e. ~P 1o A. y e. ~P 1oDECID x = y -> A. x e. ~P 1oDECID x = 1o )
11		pw1dc1 7215	. . 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 846   = wceq 1402   e. wcel 2209   A.wral 2528   ~Pcpw 3688  EXMIDwem 4329   1oc1o 6674

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-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

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-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-uni 3934  df-tr 4228  df-exmid 4330  df-iord 4509  df-on 4511  df-suc 4514  df-1o 6681

This theorem is referenced by: (None)