Theorem exmidpw 6865

Description: Excluded middle is equivalent to the power set of 1o having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)

Assertion

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

Proof of Theorem exmidpw

Step	Hyp	Ref	Expression
1		df1o2 6388	. . . . 5 |- 1o = { (/) }
2		p0ex 4161	. . . . 5 |- { (/) } e. _V
3	1, 2	eqeltri 2237	. . . 4 |- 1o e. _V
4	3	pwex 4156	. . 3 |- ~P 1o e. _V
5		exmid01 4171	. . . . . . . . 9 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
6	5	biimpi 119	. . . . . . . 8 |- (EXMID -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
7	6	19.21bi 1545	. . . . . . 7 |- (EXMID -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
8	1	pweqi 3557	. . . . . . . . 9 |- ~P 1o = ~P { (/) }
9	8	eleq2i 2231	. . . . . . . 8 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
10		velpw 3560	. . . . . . . 8 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
11	9, 10	bitri 183	. . . . . . 7 |- ( x e. ~P 1o <-> x C_ { (/) } )
12		vex 2724	. . . . . . . 8 |- x e. _V
13	12	elpr 3591	. . . . . . 7 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
14	7, 11, 13	3imtr4g 204	. . . . . 6 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
15	14	ssrdv 3143	. . . . 5 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
16		pwpw0ss 3778	. . . . . . 7 |- { (/) , { (/) } } C_ ~P { (/) }
17	16, 8	sseqtrri 3172	. . . . . 6 |- { (/) , { (/) } } C_ ~P 1o
18	17	a1i 9	. . . . 5 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
19	15, 18	eqssd 3154	. . . 4 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
20		df2o2 6390	. . . 4 |- 2o = { (/) , { (/) } }
21	19, 20	eqtr4di 2215	. . 3 |- (EXMID -> ~P 1o = 2o )
22		eqeng 6723	. . 3 |- ( ~P 1o e. _V -> ( ~P 1o = 2o -> ~P 1o ~~ 2o ) )
23	4, 21, 22	mpsyl 65	. 2 |- (EXMID -> ~P 1o ~~ 2o )
24		0nep0 4138	. . . . . . . 8 |- (/) =/= { (/) }
25		0ex 4103	. . . . . . . . . . 11 |- (/) e. _V
26	25, 2	prss 3723	. . . . . . . . . 10 |- ( ( (/) e. ~P 1o /\ { (/) } e. ~P 1o ) <-> { (/) , { (/) } } C_ ~P 1o )
27	17, 26	mpbir 145	. . . . . . . . 9 |- ( (/) e. ~P 1o /\ { (/) } e. ~P 1o )
28		en2eqpr 6864	. . . . . . . . . 10 |- ( ( ~P 1o ~~ 2o /\ (/) e. ~P 1o /\ { (/) } e. ~P 1o ) -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
29	28	3expb 1193	. . . . . . . . 9 |- ( ( ~P 1o ~~ 2o /\ ( (/) e. ~P 1o /\ { (/) } e. ~P 1o ) ) -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
30	27, 29	mpan2 422	. . . . . . . 8 |- ( ~P 1o ~~ 2o -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
31	24, 30	mpi 15	. . . . . . 7 |- ( ~P 1o ~~ 2o -> ~P 1o = { (/) , { (/) } } )
32	31	eleq2d 2234	. . . . . 6 |- ( ~P 1o ~~ 2o -> ( x e. ~P 1o <-> x e. { (/) , { (/) } } ) )
33	32, 11, 13	3bitr3g 221	. . . . 5 |- ( ~P 1o ~~ 2o -> ( x C_ { (/) } <-> ( x = (/) \/ x = { (/) } ) ) )
34	33	biimpd 143	. . . 4 |- ( ~P 1o ~~ 2o -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
35	34	alrimiv 1861	. . 3 |- ( ~P 1o ~~ 2o -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
36	35, 5	sylibr 133	. 2 |- ( ~P 1o ~~ 2o -> EXMID )
37	23, 36	impbii 125	1 |- (EXMID <-> ~P 1o ~~ 2o )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   A.wal 1340   = wceq 1342   e. wcel 2135   =/= wne 2334   _Vcvv 2721   C_ wss 3111   (/)c0 3404   ~Pcpw 3553   {csn 3570   {cpr 3571   class class class wbr 3976  EXMIDwem 4167   1oc1o 6368   2oc2o 6369   ~~ cen 6695

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-exmid 4168  df-id 4265  df-suc 4343  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-1o 6375  df-2o 6376  df-en 6698

This theorem is referenced by:  pwf1oexmid  13713