Theorem exmidpweq 6970

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 4231	. . . . . . . 8 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2	1	biimpi 120	. . . . . . 7 |- (EXMID -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
3	2	19.21bi 1572	. . . . . 6 |- (EXMID -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
4		df1o2 6487	. . . . . . . . 9 |- 1o = { (/) }
5	4	pweqi 3609	. . . . . . . 8 |- ~P 1o = ~P { (/) }
6	5	eleq2i 2263	. . . . . . 7 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
7		velpw 3612	. . . . . . 7 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
8	6, 7	bitri 184	. . . . . 6 |- ( x e. ~P 1o <-> x C_ { (/) } )
9		vex 2766	. . . . . . 7 |- x e. _V
10	9	elpr 3643	. . . . . 6 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
11	3, 8, 10	3imtr4g 205	. . . . 5 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
12	11	ssrdv 3189	. . . 4 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
13		pwpw0ss 3834	. . . . . 6 |- { (/) , { (/) } } C_ ~P { (/) }
14	13, 5	sseqtrri 3218	. . . . 5 |- { (/) , { (/) } } C_ ~P 1o
15	14	a1i 9	. . . 4 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
16	12, 15	eqssd 3200	. . 3 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
17		df2o2 6489	. . 3 |- 2o = { (/) , { (/) } }
18	16, 17	eqtr4di 2247	. 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 2290	. . . . . . 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 2245	. . . . . . 7 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ~P 1o = { (/) , { (/) } } )
24	21, 23	eleqtrd 2275	. . . . . 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 1888	. . 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 709   A.wal 1362   = wceq 1364   e. wcel 2167   C_ wss 3157   (/)c0 3450   ~Pcpw 3605   {csn 3622   {cpr 3623  EXMIDwem 4227   1oc1o 6467   2oc2o 6468

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  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-exmid 4228  df-suc 4406  df-1o 6474  df-2o 6475

This theorem is referenced by:  pw1fin  6971  pw1nel3  7298  3nsssucpw1  7303  onntri35  7304