Theorem ordpwsucexmid 4616

Description: The subset in ordpwsucss 4613 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 4207	. . . . 5 |- (/) e. ~P { z e. { (/) } | ph }
2		0elon 4437	. . . . 5 |- (/) e. On
3		elin 3355	. . . . 5 |- ( (/) e. ( ~P { z e. { (/) } | ph } i^i On ) <-> ( (/) e. ~P { z e. { (/) } | ph } /\ (/) e. On ) )
4	1, 2, 3	mpbir2an 944	. . . 4 |- (/) e. ( ~P { z e. { (/) } | ph } i^i On )
5		ordtriexmidlem 4565	. . . . 5 |- { z e. { (/) } | ph } e. On
6		suceq 4447	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> suc x = suc { z e. { (/) } | ph } )
7		pweq 3618	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ~P x = ~P { z e. { (/) } | ph } )
8	7	ineq1d 3372	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( ~P x i^i On ) = ( ~P { z e. { (/) } | ph } i^i On ) )
9	6, 8	eqeq12d 2219	. . . . . 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 2847	. . . . 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 2280	. . 3 |- (/) e. suc { z e. { (/) } | ph }
14		elsuci 4448	. . 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 4170	. . . . . 6 |- (/) e. _V
17	16	snid 3663	. . . . 5 |- (/) e. { (/) }
18		biidd 172	. . . . . 6 |- ( z = (/) -> ( ph <-> ph ) )
19	18	elrab3 2929	. . . . 5 |- ( (/) e. { (/) } -> ( (/) e. { z e. { (/) } | ph } <-> ph ) )
20	17, 19	ax-mp 5	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
21	20	biimpi 120	. . 3 |- ( (/) e. { z e. { (/) } | ph } -> ph )
22		ordtriexmidlem2 4566	. . . 4 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
23	22	eqcoms 2207	. . 3 |- ( (/) = { z e. { (/) } | ph } -> -. ph )
24	21, 23	orim12i 760	. 2 |- ( ( (/) e. { z e. { (/) } | ph } \/ (/) = { z e. { (/) } | ph } ) -> ( ph \/ -. ph ) )
25	15, 24	ax-mp 5	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   <-> wb 105   \/ wo 709   = wceq 1372   e. wcel 2175   A.wral 2483   {crab 2487   i^i cin 3164   (/)c0 3459   ~Pcpw 3615   {csn 3632   Oncon0 4408   suc csuc 4410

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-uni 3850  df-tr 4142  df-iord 4411  df-on 4413  df-suc 4416

This theorem is referenced by: (None)