Theorem pw1dc0el 6972

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 6487	. . . . . . 7 |- 1o = { (/) }
2	1	eqcomi 2200	. . . . . 6 |- { (/) } = 1o
3	2	sseq2i 3210	. . . . 5 |- ( x C_ { (/) } <-> x C_ 1o )
4		velpw 3612	. . . . 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 1484	. 2 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) <-> A. x ( x e. ~P 1o -> DECID (/) e. x ) )
8		df-exmid 4228	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
9		df-ral 2480	. 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 835   A.wal 1362   e. wcel 2167   A.wral 2475   C_ wss 3157   (/)c0 3450   ~Pcpw 3605   {csn 3622  EXMIDwem 4227   1oc1o 6467

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-exmid 4228  df-suc 4406  df-1o 6474

This theorem is referenced by:  pw1dc1  6975