Theorem pw1dc0el 7007

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 6514	. . . . . . 7 |- 1o = { (/) }
2	1	eqcomi 2208	. . . . . 6 |- { (/) } = 1o
3	2	sseq2i 3219	. . . . 5 |- ( x C_ { (/) } <-> x C_ 1o )
4		velpw 3622	. . . . 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 1492	. 2 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) <-> A. x ( x e. ~P 1o -> DECID (/) e. x ) )
8		df-exmid 4238	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
9		df-ral 2488	. 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 1370   e. wcel 2175   A.wral 2483   C_ wss 3165   (/)c0 3459   ~Pcpw 3615   {csn 3632  EXMIDwem 4237   1oc1o 6494

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-exmid 4238  df-suc 4417  df-1o 6501

This theorem is referenced by:  pw1dc1  7010