Theorem exmidpw 7066

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 6573	. . . . 5 |- 1o = { (/) }
2		p0ex 4271	. . . . 5 |- { (/) } e. _V
3	1, 2	eqeltri 2302	. . . 4 |- 1o e. _V
4	3	pwex 4266	. . 3 |- ~P 1o e. _V
5		exmid01 4281	. . . . . . . . 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 3653	. . . . . . . . 9 |- ~P 1o = ~P { (/) }
9	8	eleq2i 2296	. . . . . . . 8 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
10		velpw 3656	. . . . . . . 8 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
11	9, 10	bitri 184	. . . . . . 7 |- ( x e. ~P 1o <-> x C_ { (/) } )
12		vex 2802	. . . . . . . 8 |- x e. _V
13	12	elpr 3687	. . . . . . 7 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
14	7, 11, 13	3imtr4g 205	. . . . . 6 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
15	14	ssrdv 3230	. . . . 5 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
16		pwpw0ss 3882	. . . . . . 7 |- { (/) , { (/) } } C_ ~P { (/) }
17	16, 8	sseqtrri 3259	. . . . . 6 |- { (/) , { (/) } } C_ ~P 1o
18	17	a1i 9	. . . . 5 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
19	15, 18	eqssd 3241	. . . 4 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
20		df2o2 6575	. . . 4 |- 2o = { (/) , { (/) } }
21	19, 20	eqtr4di 2280	. . 3 |- (EXMID -> ~P 1o = 2o )
22		eqeng 6915	. . 3 |- ( ~P 1o e. _V -> ( ~P 1o = 2o -> ~P 1o ~~ 2o ) )
23	4, 21, 22	mpsyl 65	. 2 |- (EXMID -> ~P 1o ~~ 2o )
24		0nep0 4248	. . . . . . . 8 |- (/) =/= { (/) }
25		0ex 4210	. . . . . . . . . . 11 |- (/) e. _V
26	25, 2	prss 3823	. . . . . . . . . 10 |- ( ( (/) e. ~P 1o /\ { (/) } e. ~P 1o ) <-> { (/) , { (/) } } C_ ~P 1o )
27	17, 26	mpbir 146	. . . . . . . . 9 |- ( (/) e. ~P 1o /\ { (/) } e. ~P 1o )
28		en2eqpr 7065	. . . . . . . . . 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 2799   C_ wss 3197   (/)c0 3491   ~Pcpw 3649   {csn 3666   {cpr 3667   class class class wbr 4082  EXMIDwem 4277   1oc1o 6553   2oc2o 6554   ~~ cen 6883

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

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 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-exmid 4278  df-id 4383  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560  df-2o 6561  df-en 6886

This theorem is referenced by:  exmidpw2en  7070  pwf1oexmid  16324