Theorem ordpwsucexmid 4386

Description: The subset in ordpwsucss 4383 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 3999	. . . . 5 |- (/) e. ~P { z e. { (/) } | ph }
2		0elon 4219	. . . . 5 |- (/) e. On
3		elin 3183	. . . . 5 |- ( (/) e. ( ~P { z e. { (/) } | ph } i^i On ) <-> ( (/) e. ~P { z e. { (/) } | ph } /\ (/) e. On ) )
4	1, 2, 3	mpbir2an 888	. . . 4 |- (/) e. ( ~P { z e. { (/) } | ph } i^i On )
5		ordtriexmidlem 4336	. . . . 5 |- { z e. { (/) } | ph } e. On
6		suceq 4229	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> suc x = suc { z e. { (/) } | ph } )
7		pweq 3432	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ~P x = ~P { z e. { (/) } | ph } )
8	7	ineq1d 3200	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( ~P x i^i On ) = ( ~P { z e. { (/) } | ph } i^i On ) )
9	6, 8	eqeq12d 2102	. . . . . 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 2694	. . . . 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 2163	. . 3 |- (/) e. suc { z e. { (/) } | ph }
14		elsuci 4230	. . 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 3966	. . . . . 6 |- (/) e. _V
17	16	snid 3475	. . . . 5 |- (/) e. { (/) }
18		biidd 170	. . . . . 6 |- ( z = (/) -> ( ph <-> ph ) )
19	18	elrab3 2772	. . . . 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 4337	. . . 4 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
23	22	eqcoms 2091	. . 3 |- ( (/) = { z e. { (/) } | ph } -> -. ph )
24	21, 23	orim12i 711	. 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 664   = wceq 1289   e. wcel 1438   A.wral 2359   {crab 2363   i^i cin 2998   (/)c0 3286   ~Pcpw 3429   {csn 3446   Oncon0 4190   suc csuc 4192

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198

This theorem is referenced by: (None)