Theorem ordpwsucexmid 4530

Description: The subset in ordpwsucss 4527 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 4126	. . . . 5 |- (/) e. ~P { z e. { (/) } | ph }
2		0elon 4353	. . . . 5 |- (/) e. On
3		elin 3290	. . . . 5 |- ( (/) e. ( ~P { z e. { (/) } | ph } i^i On ) <-> ( (/) e. ~P { z e. { (/) } | ph } /\ (/) e. On ) )
4	1, 2, 3	mpbir2an 927	. . . 4 |- (/) e. ( ~P { z e. { (/) } | ph } i^i On )
5		ordtriexmidlem 4479	. . . . 5 |- { z e. { (/) } | ph } e. On
6		suceq 4363	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> suc x = suc { z e. { (/) } | ph } )
7		pweq 3546	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ~P x = ~P { z e. { (/) } | ph } )
8	7	ineq1d 3307	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( ~P x i^i On ) = ( ~P { z e. { (/) } | ph } i^i On ) )
9	6, 8	eqeq12d 2172	. . . . . 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 2787	. . . . 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 2233	. . 3 |- (/) e. suc { z e. { (/) } | ph }
14		elsuci 4364	. . 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 4092	. . . . . 6 |- (/) e. _V
17	16	snid 3591	. . . . 5 |- (/) e. { (/) }
18		biidd 171	. . . . . 6 |- ( z = (/) -> ( ph <-> ph ) )
19	18	elrab3 2869	. . . . 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 4480	. . . 4 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
23	22	eqcoms 2160	. . 3 |- ( (/) = { z e. { (/) } | ph } -> -. ph )
24	21, 23	orim12i 749	. 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 698   = wceq 1335   e. wcel 2128   A.wral 2435   {crab 2439   i^i cin 3101   (/)c0 3394   ~Pcpw 3543   {csn 3560   Oncon0 4324   suc csuc 4326

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-uni 3774  df-tr 4064  df-iord 4327  df-on 4329  df-suc 4332

This theorem is referenced by: (None)