Theorem ordpwsucexmid 4554

Description: The subset in ordpwsucss 4551 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 4150	. . . . 5 |- (/) e. ~P { z e. { (/) } | ph }
2		0elon 4377	. . . . 5 |- (/) e. On
3		elin 3310	. . . . 5 |- ( (/) e. ( ~P { z e. { (/) } | ph } i^i On ) <-> ( (/) e. ~P { z e. { (/) } | ph } /\ (/) e. On ) )
4	1, 2, 3	mpbir2an 937	. . . 4 |- (/) e. ( ~P { z e. { (/) } | ph } i^i On )
5		ordtriexmidlem 4503	. . . . 5 |- { z e. { (/) } | ph } e. On
6		suceq 4387	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> suc x = suc { z e. { (/) } | ph } )
7		pweq 3569	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ~P x = ~P { z e. { (/) } | ph } )
8	7	ineq1d 3327	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( ~P x i^i On ) = ( ~P { z e. { (/) } | ph } i^i On ) )
9	6, 8	eqeq12d 2185	. . . . . 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 2805	. . . . 5 |- ( { z e. { (/) } | ph } e. On -> suc { z e. { (/) } | ph } = ( ~P { z e. { (/) } | ph } i^i On ) )
12	5, 11	ax-mp 5	. . . 4 |- suc { z e. { (/) } | ph } = ( ~P { z e. { (/) } | ph } i^i On )
13	4, 12	eleqtrri 2246	. . 3 |- (/) e. suc { z e. { (/) } | ph }
14		elsuci 4388	. . 3 |- ( (/) e. suc { z e. { (/) } | ph } -> ( (/) e. { z e. { (/) } | ph } \/ (/) = { z e. { (/) } | ph } ) )
15	13, 14	ax-mp 5	. 2 |- ( (/) e. { z e. { (/) } | ph } \/ (/) = { z e. { (/) } | ph } )
16		0ex 4116	. . . . . 6 |- (/) e. _V
17	16	snid 3614	. . . . 5 |- (/) e. { (/) }
18		biidd 171	. . . . . 6 |- ( z = (/) -> ( ph <-> ph ) )
19	18	elrab3 2887	. . . . 5 |- ( (/) e. { (/) } -> ( (/) e. { z e. { (/) } | ph } <-> ph ) )
20	17, 19	ax-mp 5	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
21	20	biimpi 119	. . 3 |- ( (/) e. { z e. { (/) } | ph } -> ph )
22		ordtriexmidlem2 4504	. . . 4 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
23	22	eqcoms 2173	. . 3 |- ( (/) = { z e. { (/) } | ph } -> -. ph )
24	21, 23	orim12i 754	. 2 |- ( ( (/) e. { z e. { (/) } | ph } \/ (/) = { z e. { (/) } | ph } ) -> ( ph \/ -. ph ) )
25	15, 24	ax-mp 5	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   A.wral 2448   {crab 2452   i^i cin 3120   (/)c0 3414   ~Pcpw 3566   {csn 3583   Oncon0 4348   suc csuc 4350

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356

This theorem is referenced by: (None)