Theorem ordpwsucexmid 4313

Description: The subset in ordpwsucss 4310 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)

Hypothesis

Ref	Expression
ordpwsucexmid.1	|- A. x e. On suc x = ( ~P x i^i On )

Assertion

Ref	Expression
ordpwsucexmid	|- ( ph \/ -. ph )

Distinct variable group:   ph, x

Proof of Theorem ordpwsucexmid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elpw 3938	. . . . 5 |- (/) e. ~P { z e. { (/) } | ph }
2		0elon 4147	. . . . 5 |- (/) e. On
3		elin 3155	. . . . 5 |- ( (/) e. ( ~P { z e. { (/) } | ph } i^i On ) <-> ( (/) e. ~P { z e. { (/) } | ph } /\ (/) e. On ) )
4	1, 2, 3	mpbir2an 883	. . . 4 |- (/) e. ( ~P { z e. { (/) } | ph } i^i On )
5		ordtriexmidlem 4263	. . . . 5 |- { z e. { (/) } | ph } e. On
6		suceq 4157	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> suc x = suc { z e. { (/) } | ph } )
7		pweq 3385	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ~P x = ~P { z e. { (/) } | ph } )
8	7	ineq1d 3166	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( ~P x i^i On ) = ( ~P { z e. { (/) } | ph } i^i On ) )
9	6, 8	eqeq12d 2095	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> ( suc x = ( ~P x i^i On ) <-> suc { z e. { (/) } | ph } = ( ~P { z e. { (/) } | ph } i^i On ) ) )
10		ordpwsucexmid.1	. . . . . 6 |- A. x e. On suc x = ( ~P x i^i On )
11	9, 10	vtoclri 2673	. . . . 5 |- ( { z e. { (/) } | ph } e. On -> suc { z e. { (/) } | ph } = ( ~P { z e. { (/) } | ph } i^i On ) )
12	5, 11	ax-mp 7	. . . 4 |- suc { z e. { (/) } | ph } = ( ~P { z e. { (/) } | ph } i^i On )
13	4, 12	eleqtrri 2154	. . 3 |- (/) e. suc { z e. { (/) } | ph }
14		elsuci 4158	. . 3 |- ( (/) e. suc { z e. { (/) } | ph } -> ( (/) e. { z e. { (/) } | ph } \/ (/) = { z e. { (/) } | ph } ) )
15	13, 14	ax-mp 7	. 2 |- ( (/) e. { z e. { (/) } | ph } \/ (/) = { z e. { (/) } | ph } )
16		0ex 3905	. . . . . 6 |- (/) e. _V
17	16	snid 3425	. . . . 5 |- (/) e. { (/) }
18		biidd 170	. . . . . 6 |- ( z = (/) -> ( ph <-> ph ) )
19	18	elrab3 2750	. . . . 5 |- ( (/) e. { (/) } -> ( (/) e. { z e. { (/) } | ph } <-> ph ) )
20	17, 19	ax-mp 7	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
21	20	biimpi 118	. . 3 |- ( (/) e. { z e. { (/) } | ph } -> ph )
22		ordtriexmidlem2 4264	. . . 4 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
23	22	eqcoms 2084	. . 3 |- ( (/) = { z e. { (/) } | ph } -> -. ph )
24	21, 23	orim12i 708	. 2 |- ( ( (/) e. { z e. { (/) } | ph } \/ (/) = { z e. { (/) } | ph } ) -> ( ph \/ -. ph ) )
25	15, 24	ax-mp 7	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   A.wral 2348   {crab 2352   i^i cin 2972   (/)c0 3251   ~Pcpw 3382   {csn 3398   Oncon0 4118   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126

This theorem is referenced by: (None)