Theorem pw1dc0el 6910

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 6429	. . . . . . 7 |- 1o = { (/) }
2	1	eqcomi 2181	. . . . . 6 |- { (/) } = 1o
3	2	sseq2i 3182	. . . . 5 |- ( x C_ { (/) } <-> x C_ 1o )
4		velpw 3582	. . . . 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 1470	. 2 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) <-> A. x ( x e. ~P 1o -> DECID (/) e. x ) )
8		df-exmid 4195	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
9		df-ral 2460	. 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 834   A.wal 1351   e. wcel 2148   A.wral 2455   C_ wss 3129   (/)c0 3422   ~Pcpw 3575   {csn 3592  EXMIDwem 4194   1oc1o 6409

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-exmid 4195  df-suc 4371  df-1o 6416

This theorem is referenced by:  pw1dc1  6912