Theorem exmidpw 7020

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 6528	. . . . 5 |- 1o = { (/) }
2		p0ex 4240	. . . . 5 |- { (/) } e. _V
3	1, 2	eqeltri 2279	. . . 4 |- 1o e. _V
4	3	pwex 4235	. . 3 |- ~P 1o e. _V
5		exmid01 4250	. . . . . . . . 9 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
6	5	biimpi 120	. . . . . . . 8 |- (EXMID -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
7	6	19.21bi 1582	. . . . . . 7 |- (EXMID -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
8	1	pweqi 3625	. . . . . . . . 9 |- ~P 1o = ~P { (/) }
9	8	eleq2i 2273	. . . . . . . 8 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
10		velpw 3628	. . . . . . . 8 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
11	9, 10	bitri 184	. . . . . . 7 |- ( x e. ~P 1o <-> x C_ { (/) } )
12		vex 2776	. . . . . . . 8 |- x e. _V
13	12	elpr 3659	. . . . . . 7 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
14	7, 11, 13	3imtr4g 205	. . . . . 6 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
15	14	ssrdv 3203	. . . . 5 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
16		pwpw0ss 3851	. . . . . . 7 |- { (/) , { (/) } } C_ ~P { (/) }
17	16, 8	sseqtrri 3232	. . . . . 6 |- { (/) , { (/) } } C_ ~P 1o
18	17	a1i 9	. . . . 5 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
19	15, 18	eqssd 3214	. . . 4 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
20		df2o2 6530	. . . 4 |- 2o = { (/) , { (/) } }
21	19, 20	eqtr4di 2257	. . 3 |- (EXMID -> ~P 1o = 2o )
22		eqeng 6870	. . 3 |- ( ~P 1o e. _V -> ( ~P 1o = 2o -> ~P 1o ~~ 2o ) )
23	4, 21, 22	mpsyl 65	. 2 |- (EXMID -> ~P 1o ~~ 2o )
24		0nep0 4217	. . . . . . . 8 |- (/) =/= { (/) }
25		0ex 4179	. . . . . . . . . . 11 |- (/) e. _V
26	25, 2	prss 3795	. . . . . . . . . 10 |- ( ( (/) e. ~P 1o /\ { (/) } e. ~P 1o ) <-> { (/) , { (/) } } C_ ~P 1o )
27	17, 26	mpbir 146	. . . . . . . . 9 |- ( (/) e. ~P 1o /\ { (/) } e. ~P 1o )
28		en2eqpr 7019	. . . . . . . . . 10 |- ( ( ~P 1o ~~ 2o /\ (/) e. ~P 1o /\ { (/) } e. ~P 1o ) -> ( (/) =/= { (/) } -> ~P 1o = { (/) , { (/) } } ) )
29	28	3expb 1207	. . . . . . . . 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 2276	. . . . . 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 1898	. . 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 710   A.wal 1371   = wceq 1373   e. wcel 2177   =/= wne 2377   _Vcvv 2773   C_ wss 3170   (/)c0 3464   ~Pcpw 3621   {csn 3638   {cpr 3639   class class class wbr 4051  EXMIDwem 4246   1oc1o 6508   2oc2o 6509   ~~ cen 6838

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-exmid 4247  df-id 4348  df-suc 4426  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-1o 6515  df-2o 6516  df-en 6841

This theorem is referenced by:  exmidpw2en  7024  pwf1oexmid  16077