Theorem pw1dc0el 6853

Description: Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)

Assertion

Ref	Expression
pw1dc0el	|- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )

Proof of Theorem pw1dc0el

Step	Hyp	Ref	Expression
1		df1o2 6373	. . . . . . 7 |- 1o = { (/) }
2	1	eqcomi 2161	. . . . . 6 |- { (/) } = 1o
3	2	sseq2i 3155	. . . . 5 |- ( x C_ { (/) } <-> x C_ 1o )
4		velpw 3550	. . . . 5 |- ( x e. ~P 1o <-> x C_ 1o )
5	3, 4	bitr4i 186	. . . 4 |- ( x C_ { (/) } <-> x e. ~P 1o )
6	5	imbi1i 237	. . 3 |- ( ( x C_ { (/) } -> DECID (/) e. x ) <-> ( x e. ~P 1o -> DECID (/) e. x ) )
7	6	albii 1450	. 2 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) <-> A. x ( x e. ~P 1o -> DECID (/) e. x ) )
8		df-exmid 4156	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
9		df-ral 2440	. 2 |- ( A. x e. ~P 1oDECID (/) e. x <-> A. x ( x e. ~P 1o -> DECID (/) e. x ) )
10	7, 8, 9	3bitr4i 211	1 |- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )

Syntax hints:   -> wi 4   <-> wb 104  DECID wdc 820   A.wal 1333   e. wcel 2128   A.wral 2435   C_ wss 3102   (/)c0 3394   ~Pcpw 3543   {csn 3560  EXMIDwem 4155   1oc1o 6353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-exmid 4156  df-suc 4331  df-1o 6360

This theorem is referenced by:  pw1dc1  6855