Theorem exmidpw 6802

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 6326	. . . . 5 |- 1o = { (/) }
2		p0ex 4112	. . . . 5 |- { (/) } e. _V
3	1, 2	eqeltri 2212	. . . 4 |- 1o e. _V
4	3	pwex 4107	. . 3 |- ~P 1o e. _V
5		exmid01 4121	. . . . . . . . 9 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
6	5	biimpi 119	. . . . . . . 8 |- (EXMID -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
7	6	19.21bi 1537	. . . . . . 7 |- (EXMID -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
8	1	pweqi 3514	. . . . . . . . 9 |- ~P 1o = ~P { (/) }
9	8	eleq2i 2206	. . . . . . . 8 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
10		velpw 3517	. . . . . . . 8 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
11	9, 10	bitri 183	. . . . . . 7 |- ( x e. ~P 1o <-> x C_ { (/) } )
12		vex 2689	. . . . . . . 8 |- x e. _V
13	12	elpr 3548	. . . . . . 7 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
14	7, 11, 13	3imtr4g 204	. . . . . 6 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
15	14	ssrdv 3103	. . . . 5 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
16		pwpw0ss 3731	. . . . . . 7 |- { (/) , { (/) } } C_ ~P { (/) }
17	16, 8	sseqtrri 3132	. . . . . 6 |- { (/) , { (/) } } C_ ~P 1o
18	17	a1i 9	. . . . 5 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
19	15, 18	eqssd 3114	. . . 4 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
20		df2o2 6328	. . . 4 |- 2o = { (/) , { (/) } }
21	19, 20	syl6eqr 2190	. . 3 |- (EXMID -> ~P 1o = 2o )
22		eqeng 6660	. . 3 |- ( ~P 1o e. _V -> ( ~P 1o = 2o -> ~P 1o ~~ 2o ) )
23	4, 21, 22	mpsyl 65	. 2 |- (EXMID -> ~P 1o ~~ 2o )
24		0nep0 4089	. . . . . . . 8 |- (/) =/= { (/) }
25		0ex 4055	. . . . . . . . . . 11 |- (/) e. _V
26	25, 2	prss 3676	. . . . . . . . . 10 |- ( ( (/) e. ~P 1o /\ { (/) } e. ~P 1o ) <-> { (/) , { (/) } } C_ ~P 1o )
27	17, 26	mpbir 145	. . . . . . . . 9 |- ( (/) e. ~P 1o /\ { (/) } e. ~P 1o )
28		en2eqpr 6801	. . . . . . . . . 10 |- ( ( ~P 1o ~~ 2o /\ (/) e. ~P 1o /\ { (/) } e. ~P 1o ) -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
29	28	3expb 1182	. . . . . . . . 9 |- ( ( ~P 1o ~~ 2o /\ ( (/) e. ~P 1o /\ { (/) } e. ~P 1o ) ) -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
30	27, 29	mpan2 421	. . . . . . . 8 |- ( ~P 1o ~~ 2o -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
31	24, 30	mpi 15	. . . . . . 7 |- ( ~P 1o ~~ 2o -> ~P 1o = { (/) , { (/) } } )
32	31	eleq2d 2209	. . . . . 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 1846	. . 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 697   A.wal 1329   = wceq 1331   e. wcel 1480   =/= wne 2308   _Vcvv 2686   C_ wss 3071   (/)c0 3363   ~Pcpw 3510   {csn 3527   {cpr 3528   class class class wbr 3929  EXMIDwem 4118   1oc1o 6306   2oc2o 6307   ~~ cen 6632

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-exmid 4119  df-id 4215  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1o 6313  df-2o 6314  df-en 6635

This theorem is referenced by:  pwf1oexmid  13194