Theorem exmidpweq 6875

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 4177	. . . . . . . 8 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2	1	biimpi 119	. . . . . . 7 |- (EXMID -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
3	2	19.21bi 1546	. . . . . 6 |- (EXMID -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
4		df1o2 6397	. . . . . . . . 9 |- 1o = { (/) }
5	4	pweqi 3563	. . . . . . . 8 |- ~P 1o = ~P { (/) }
6	5	eleq2i 2233	. . . . . . 7 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
7		velpw 3566	. . . . . . 7 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
8	6, 7	bitri 183	. . . . . 6 |- ( x e. ~P 1o <-> x C_ { (/) } )
9		vex 2729	. . . . . . 7 |- x e. _V
10	9	elpr 3597	. . . . . 6 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
11	3, 8, 10	3imtr4g 204	. . . . 5 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
12	11	ssrdv 3148	. . . 4 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
13		pwpw0ss 3784	. . . . . 6 |- { (/) , { (/) } } C_ ~P { (/) }
14	13, 5	sseqtrri 3177	. . . . 5 |- { (/) , { (/) } } C_ ~P 1o
15	14	a1i 9	. . . 4 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
16	12, 15	eqssd 3159	. . 3 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
17		df2o2 6399	. . 3 |- 2o = { (/) , { (/) } }
18	16, 17	eqtr4di 2217	. 2 |- (EXMID -> ~P 1o = 2o )
19		simpr 109	. . . . . . . . 9 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x C_ { (/) } )
20	19, 7	sylibr 133	. . . . . . . 8 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x e. ~P { (/) } )
21	20, 5	eleqtrrdi 2260	. . . . . . 7 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x e. ~P 1o )
22		simpl 108	. . . . . . . 8 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ~P 1o = 2o )
23	22, 17	eqtrdi 2215	. . . . . . 7 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ~P 1o = { (/) , { (/) } } )
24	21, 23	eleqtrd 2245	. . . . . 6 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x e. { (/) , { (/) } } )
25	24, 10	sylib 121	. . . . 5 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ( x = (/) \/ x = { (/) } ) )
26	25	ex 114	. . . 4 |- ( ~P 1o = 2o -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
27	26	alrimiv 1862	. . 3 |- ( ~P 1o = 2o -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
28	27, 1	sylibr 133	. 2 |- ( ~P 1o = 2o -> EXMID )
29	18, 28	impbii 125	1 |- (EXMID <-> ~P 1o = 2o )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   A.wal 1341   = wceq 1343   e. wcel 2136   C_ wss 3116   (/)c0 3409   ~Pcpw 3559   {csn 3576   {cpr 3577  EXMIDwem 4173   1oc1o 6377   2oc2o 6378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-exmid 4174  df-suc 4349  df-1o 6384  df-2o 6385

This theorem is referenced by:  pw1fin  6876  pw1nel3  7187  3nsssucpw1  7192  onntri35  7193