Theorem exmidpweq 7144

Description: Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.)

Assertion

Ref	Expression
exmidpweq	|- (EXMID <-> ~P 1o = 2o )

Proof of Theorem exmidpweq

Step	Hyp	Ref	Expression
1		exmid01 4294	. . . . . . . 8 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2	1	biimpi 120	. . . . . . 7 |- (EXMID -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
3	2	19.21bi 1607	. . . . . 6 |- (EXMID -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
4		df1o2 6639	. . . . . . . . 9 |- 1o = { (/) }
5	4	pweqi 3660	. . . . . . . 8 |- ~P 1o = ~P { (/) }
6	5	eleq2i 2298	. . . . . . 7 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
7		velpw 3663	. . . . . . 7 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
8	6, 7	bitri 184	. . . . . 6 |- ( x e. ~P 1o <-> x C_ { (/) } )
9		vex 2806	. . . . . . 7 |- x e. _V
10	9	elpr 3694	. . . . . 6 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
11	3, 8, 10	3imtr4g 205	. . . . 5 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
12	11	ssrdv 3234	. . . 4 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
13		pwpw0ss 3893	. . . . . 6 |- { (/) , { (/) } } C_ ~P { (/) }
14	13, 5	sseqtrri 3263	. . . . 5 |- { (/) , { (/) } } C_ ~P 1o
15	14	a1i 9	. . . 4 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
16	12, 15	eqssd 3245	. . 3 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
17		df2o2 6641	. . 3 |- 2o = { (/) , { (/) } }
18	16, 17	eqtr4di 2282	. 2 |- (EXMID -> ~P 1o = 2o )
19		simpr 110	. . . . . . . . 9 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x C_ { (/) } )
20	19, 7	sylibr 134	. . . . . . . 8 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x e. ~P { (/) } )
21	20, 5	eleqtrrdi 2325	. . . . . . 7 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x e. ~P 1o )
22		simpl 109	. . . . . . . 8 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ~P 1o = 2o )
23	22, 17	eqtrdi 2280	. . . . . . 7 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ~P 1o = { (/) , { (/) } } )
24	21, 23	eleqtrd 2310	. . . . . 6 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x e. { (/) , { (/) } } )
25	24, 10	sylib 122	. . . . 5 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ( x = (/) \/ x = { (/) } ) )
26	25	ex 115	. . . 4 |- ( ~P 1o = 2o -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
27	26	alrimiv 1922	. . 3 |- ( ~P 1o = 2o -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
28	27, 1	sylibr 134	. 2 |- ( ~P 1o = 2o -> EXMID )
29	18, 28	impbii 126	1 |- (EXMID <-> ~P 1o = 2o )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   A.wal 1396   = wceq 1398   e. wcel 2202   C_ wss 3201   (/)c0 3496   ~Pcpw 3656   {csn 3673   {cpr 3674  EXMIDwem 4290   1oc1o 6618   2oc2o 6619

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 2213  ax-nul 4220

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-exmid 4291  df-suc 4474  df-1o 6625  df-2o 6626

This theorem is referenced by:  pw1fin  7145  pw1nel3  7509  3nsssucpw1  7514  onntri35  7515