Theorem exmidpw 7093

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 6591	. . . . 5 |- 1o = { (/) }
2		p0ex 4276	. . . . 5 |- { (/) } e. _V
3	1, 2	eqeltri 2302	. . . 4 |- 1o e. _V
4	3	pwex 4271	. . 3 |- ~P 1o e. _V
5		exmid01 4286	. . . . . . . . 9 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
6	5	biimpi 120	. . . . . . . 8 |- (EXMID -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
7	6	19.21bi 1604	. . . . . . 7 |- (EXMID -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
8	1	pweqi 3654	. . . . . . . . 9 |- ~P 1o = ~P { (/) }
9	8	eleq2i 2296	. . . . . . . 8 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
10		velpw 3657	. . . . . . . 8 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
11	9, 10	bitri 184	. . . . . . 7 |- ( x e. ~P 1o <-> x C_ { (/) } )
12		vex 2803	. . . . . . . 8 |- x e. _V
13	12	elpr 3688	. . . . . . 7 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
14	7, 11, 13	3imtr4g 205	. . . . . 6 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
15	14	ssrdv 3231	. . . . 5 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
16		pwpw0ss 3886	. . . . . . 7 |- { (/) , { (/) } } C_ ~P { (/) }
17	16, 8	sseqtrri 3260	. . . . . 6 |- { (/) , { (/) } } C_ ~P 1o
18	17	a1i 9	. . . . 5 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
19	15, 18	eqssd 3242	. . . 4 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
20		df2o2 6593	. . . 4 |- 2o = { (/) , { (/) } }
21	19, 20	eqtr4di 2280	. . 3 |- (EXMID -> ~P 1o = 2o )
22		eqeng 6934	. . 3 |- ( ~P 1o e. _V -> ( ~P 1o = 2o -> ~P 1o ~~ 2o ) )
23	4, 21, 22	mpsyl 65	. 2 |- (EXMID -> ~P 1o ~~ 2o )
24		0nep0 4253	. . . . . . . 8 |- (/) =/= { (/) }
25		0ex 4214	. . . . . . . . . . 11 |- (/) e. _V
26	25, 2	prss 3827	. . . . . . . . . 10 |- ( ( (/) e. ~P 1o /\ { (/) } e. ~P 1o ) <-> { (/) , { (/) } } C_ ~P 1o )
27	17, 26	mpbir 146	. . . . . . . . 9 |- ( (/) e. ~P 1o /\ { (/) } e. ~P 1o )
28		en2eqpr 7092	. . . . . . . . . 10 |- ( ( ~P 1o ~~ 2o /\ (/) e. ~P 1o /\ { (/) } e. ~P 1o ) -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
29	28	3expb 1228	. . . . . . . . 9 |- ( ( ~P 1o ~~ 2o /\ ( (/) e. ~P 1o /\ { (/) } e. ~P 1o ) ) -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
30	27, 29	mpan2 425	. . . . . . . 8 |- ( ~P 1o ~~ 2o -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
31	24, 30	mpi 15	. . . . . . 7 |- ( ~P 1o ~~ 2o -> ~P 1o = { (/) , { (/) } } )
32	31	eleq2d 2299	. . . . . 6 |- ( ~P 1o ~~ 2o -> ( x e. ~P 1o <-> x e. { (/) , { (/) } } ) )
33	32, 11, 13	3bitr3g 222	. . . . 5 |- ( ~P 1o ~~ 2o -> ( x C_ { (/) } <-> ( x = (/) \/ x = { (/) } ) ) )
34	33	biimpd 144	. . . 4 |- ( ~P 1o ~~ 2o -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
35	34	alrimiv 1920	. . 3 |- ( ~P 1o ~~ 2o -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
36	35, 5	sylibr 134	. 2 |- ( ~P 1o ~~ 2o -> EXMID )
37	23, 36	impbii 126	1 |- (EXMID <-> ~P 1o ~~ 2o )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   A.wal 1393   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2800   C_ wss 3198   (/)c0 3492   ~Pcpw 3650   {csn 3667   {cpr 3668   class class class wbr 4086  EXMIDwem 4282   1oc1o 6570   2oc2o 6571   ~~ cen 6902

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-exmid 4283  df-id 4388  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-2o 6578  df-en 6905

This theorem is referenced by:  exmidpw2en  7097  pwf1oexmid  16536