Theorem pw1dc0el 7184

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 6674	. . . . . . 7 |- 1o = { (/) }
2	1	eqcomi 2238	. . . . . 6 |- { (/) } = 1o
3	2	sseq2i 3269	. . . . 5 |- ( x C_ { (/) } <-> x C_ 1o )
4		velpw 3681	. . . . 5 |- ( x e. ~P 1o <-> x C_ 1o )
5	3, 4	bitr4i 187	. . . 4 |- ( x C_ { (/) } <-> x e. ~P 1o )
6	5	imbi1i 238	. . 3 |- ( ( x C_ { (/) } -> DECID (/) e. x ) <-> ( x e. ~P 1o -> DECID (/) e. x ) )
7	6	albii 1519	. 2 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) <-> A. x ( x e. ~P 1o -> DECID (/) e. x ) )
8		df-exmid 4313	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
9		df-ral 2527	. 2 |- ( A. x e. ~P 1oDECID (/) e. x <-> A. x ( x e. ~P 1o -> DECID (/) e. x ) )
10	7, 8, 9	3bitr4i 212	1 |- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )

Syntax hints:   -> wi 4   <-> wb 105  DECID wdc 842   A.wal 1396   e. wcel 2205   A.wral 2522   C_ wss 3214   (/)c0 3512   ~Pcpw 3674   {csn 3694  EXMIDwem 4312   1oc1o 6653

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-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-exmid 4313  df-suc 4497  df-1o 6660

This theorem is referenced by:  pw1dc1  7187