Theorem onsucsssucexmid 4625

Description: The converse of onsucsssucr 4607 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)

Hypothesis

Ref	Expression
onsucsssucexmid.1	|- A. x e. On A. y e. On ( x C_ y -> suc x C_ suc y )

Assertion

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

Distinct variable groups:   ph, x   x, y

Allowed substitution hint:   ph( y)

Proof of Theorem onsucsssucexmid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3312	. . . . . 6 |- { z e. { (/) } | ph } C_ { (/) }
2		ordtriexmidlem 4617	. . . . . . 7 |- { z e. { (/) } | ph } e. On
3		sseq1 3250	. . . . . . . . 9 |- ( x = { z e. { (/) } | ph } -> ( x C_ { (/) } <-> { z e. { (/) } | ph } C_ { (/) } ) )
4		suceq 4499	. . . . . . . . . 10 |- ( x = { z e. { (/) } | ph } -> suc x = suc { z e. { (/) } | ph } )
5	4	sseq1d 3256	. . . . . . . . 9 |- ( x = { z e. { (/) } | ph } -> ( suc x C_ suc { (/) } <-> suc { z e. { (/) } | ph } C_ suc { (/) } ) )
6	3, 5	imbi12d 234	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ( ( x C_ { (/) } -> suc x C_ suc { (/) } ) <-> ( { z e. { (/) } | ph } C_ { (/) } -> suc { z e. { (/) } | ph } C_ suc { (/) } ) ) )
7		suc0 4508	. . . . . . . . . 10 |- suc (/) = { (/) }
8		0elon 4489	. . . . . . . . . . 11 |- (/) e. On
9	8	onsuci 4614	. . . . . . . . . 10 |- suc (/) e. On
10	7, 9	eqeltrri 2305	. . . . . . . . 9 |- { (/) } e. On
11		p0ex 4278	. . . . . . . . . 10 |- { (/) } e. _V
12		eleq1 2294	. . . . . . . . . . . 12 |- ( y = { (/) } -> ( y e. On <-> { (/) } e. On ) )
13	12	anbi2d 464	. . . . . . . . . . 11 |- ( y = { (/) } -> ( ( x e. On /\ y e. On ) <-> ( x e. On /\ { (/) } e. On ) ) )
14		sseq2 3251	. . . . . . . . . . . 12 |- ( y = { (/) } -> ( x C_ y <-> x C_ { (/) } ) )
15		suceq 4499	. . . . . . . . . . . . 13 |- ( y = { (/) } -> suc y = suc { (/) } )
16	15	sseq2d 3257	. . . . . . . . . . . 12 |- ( y = { (/) } -> ( suc x C_ suc y <-> suc x C_ suc { (/) } ) )
17	14, 16	imbi12d 234	. . . . . . . . . . 11 |- ( y = { (/) } -> ( ( x C_ y -> suc x C_ suc y ) <-> ( x C_ { (/) } -> suc x C_ suc { (/) } ) ) )
18	13, 17	imbi12d 234	. . . . . . . . . 10 |- ( y = { (/) } -> ( ( ( x e. On /\ y e. On ) -> ( x C_ y -> suc x C_ suc y ) ) <-> ( ( x e. On /\ { (/) } e. On ) -> ( x C_ { (/) } -> suc x C_ suc { (/) } ) ) ) )
19		onsucsssucexmid.1	. . . . . . . . . . 11 |- A. x e. On A. y e. On ( x C_ y -> suc x C_ suc y )
20	19	rspec2 2621	. . . . . . . . . 10 |- ( ( x e. On /\ y e. On ) -> ( x C_ y -> suc x C_ suc y ) )
21	11, 18, 20	vtocl 2858	. . . . . . . . 9 |- ( ( x e. On /\ { (/) } e. On ) -> ( x C_ { (/) } -> suc x C_ suc { (/) } ) )
22	10, 21	mpan2 425	. . . . . . . 8 |- ( x e. On -> ( x C_ { (/) } -> suc x C_ suc { (/) } ) )
23	6, 22	vtoclga 2870	. . . . . . 7 |- ( { z e. { (/) } | ph } e. On -> ( { z e. { (/) } | ph } C_ { (/) } -> suc { z e. { (/) } | ph } C_ suc { (/) } ) )
24	2, 23	ax-mp 5	. . . . . 6 |- ( { z e. { (/) } | ph } C_ { (/) } -> suc { z e. { (/) } | ph } C_ suc { (/) } )
25	1, 24	ax-mp 5	. . . . 5 |- suc { z e. { (/) } | ph } C_ suc { (/) }
26	10	onsuci 4614	. . . . . . 7 |- suc { (/) } e. On
27	26	onordi 4523	. . . . . 6 |- Ord suc { (/) }
28		ordelsuc 4603	. . . . . 6 |- ( ( { z e. { (/) } | ph } e. On /\ Ord suc { (/) } ) -> ( { z e. { (/) } | ph } e. suc { (/) } <-> suc { z e. { (/) } | ph } C_ suc { (/) } ) )
29	2, 27, 28	mp2an 426	. . . . 5 |- ( { z e. { (/) } | ph } e. suc { (/) } <-> suc { z e. { (/) } | ph } C_ suc { (/) } )
30	25, 29	mpbir 146	. . . 4 |- { z e. { (/) } | ph } e. suc { (/) }
31		elsucg 4501	. . . . 5 |- ( { z e. { (/) } | ph } e. On -> ( { z e. { (/) } | ph } e. suc { (/) } <-> ( { z e. { (/) } | ph } e. { (/) } \/ { z e. { (/) } | ph } = { (/) } ) ) )
32	2, 31	ax-mp 5	. . . 4 |- ( { z e. { (/) } | ph } e. suc { (/) } <-> ( { z e. { (/) } | ph } e. { (/) } \/ { z e. { (/) } | ph } = { (/) } ) )
33	30, 32	mpbi 145	. . 3 |- ( { z e. { (/) } | ph } e. { (/) } \/ { z e. { (/) } | ph } = { (/) } )
34		elsni 3687	. . . . 5 |- ( { z e. { (/) } | ph } e. { (/) } -> { z e. { (/) } | ph } = (/) )
35		ordtriexmidlem2 4618	. . . . 5 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
36	34, 35	syl 14	. . . 4 |- ( { z e. { (/) } | ph } e. { (/) } -> -. ph )
37		0ex 4216	. . . . 5 |- (/) e. _V
38		biidd 172	. . . . 5 |- ( z = (/) -> ( ph <-> ph ) )
39	37, 38	rabsnt 3746	. . . 4 |- ( { z e. { (/) } | ph } = { (/) } -> ph )
40	36, 39	orim12i 766	. . 3 |- ( ( { z e. { (/) } | ph } e. { (/) } \/ { z e. { (/) } | ph } = { (/) } ) -> ( -. ph \/ ph ) )
41	33, 40	ax-mp 5	. 2 |- ( -. ph \/ ph )
42		orcom 735	. 2 |- ( ( -. ph \/ ph ) <-> ( ph \/ -. ph ) )
43	41, 42	mpbi 145	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   A.wral 2510   {crab 2514   C_ wss 3200   (/)c0 3494   {csn 3669   Ord word 4459   Oncon0 4460   suc csuc 4462

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468

This theorem is referenced by:  oawordriexmid  6637