Theorem onsucsssucexmid 4196

Description: The converse of onsucsssucr 4184 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 3002	. . . . . 6 |- {z e. { (/) } | ph } C_ { (/) }
2		ordtriexmidlem 4192	. . . . . . 7 |- {z e. { (/) } | ph } e. On
3		sseq1 2943	. . . . . . . . 9 |- (x = {z e. { (/) } | ph } -> (x C_ { (/) } <-> {z e. { (/) } | ph } C_ { (/) }))
4		suceq 4088	. . . . . . . . . 10 |- (x = {z e. { (/) } | ph } -> suc x = suc {z e. { (/) } | ph })
5	4	sseq1d 2949	. . . . . . . . 9 |- (x = {z e. { (/) } | ph } -> ( suc x C_ suc { (/) } <-> suc {z e. { (/) } | ph } C_ suc { (/) }))
6	3, 5	imbi12d 223	. . . . . . . 8 |- (x = {z e. { (/) } | ph } -> ((x C_ { (/) } -> suc x C_ suc { (/) }) <-> ( {z e. { (/) } | ph } C_ { (/) } -> suc {z e. { (/) } | ph } C_ suc { (/) })))
7		suc0 4097	. . . . . . . . . 10 |- suc (/) = { (/) }
8		0elon 4078	. . . . . . . . . . 11 |- (/) e. On
9	8	onsuci 4191	. . . . . . . . . 10 |- suc (/) e. On
10	7, 9	eqeltrri 2093	. . . . . . . . 9 |- { (/) } e. On
11		p0ex 3913	. . . . . . . . . 10 |- { (/) } e. _V
12		eleq1 2082	. . . . . . . . . . . 12 |- (y = { (/) } -> (y e. On <-> { (/) } e. On))
13	12	anbi2d 440	. . . . . . . . . . 11 |- (y = { (/) } -> ((x e. On /\ y e. On) <-> (x e. On /\ { (/) } e. On)))
14		sseq2 2944	. . . . . . . . . . . 12 |- (y = { (/) } -> (x C_ y <-> x C_ { (/) }))
15		suceq 4088	. . . . . . . . . . . . 13 |- (y = { (/) } -> suc y = suc { (/) })
16	15	sseq2d 2950	. . . . . . . . . . . 12 |- (y = { (/) } -> ( suc x C_ suc y <-> suc x C_ suc { (/) }))
17	14, 16	imbi12d 223	. . . . . . . . . . 11 |- (y = { (/) } -> ((x C_ y -> suc x C_ suc y) <-> (x C_ { (/) } -> suc x C_ suc { (/) })))
18	13, 17	imbi12d 223	. . . . . . . . . 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 2386	. . . . . . . . . 10 |- ((x e. On /\ y e. On) -> (x C_ y -> suc x C_ suc y))
21	11, 18, 20	vtocl 2585	. . . . . . . . 9 |- ((x e. On /\ { (/) } e. On) -> (x C_ { (/) } -> suc x C_ suc { (/) }))
22	10, 21	mpan2 403	. . . . . . . 8 |- (x e. On -> (x C_ { (/) } -> suc x C_ suc { (/) }))
23	6, 22	vtoclga 2596	. . . . . . 7 |- ( {z e. { (/) } | ph } e. On -> ( {z e. { (/) } | ph } C_ { (/) } -> suc {z e. { (/) } | ph } C_ suc { (/) }))
24	2, 23	ax-mp 7	. . . . . 6 |- ( {z e. { (/) } | ph } C_ { (/) } -> suc {z e. { (/) } | ph } C_ suc { (/) })
25	1, 24	ax-mp 7	. . . . 5 |- suc {z e. { (/) } | ph } C_ suc { (/) }
26	10	onsuci 4191	. . . . . . 7 |- suc { (/) } e. On
27	26	onordi 4113	. . . . . 6 |- Ord suc { (/) }
28		ordelsuc 4181	. . . . . 6 |- (( {z e. { (/) } | ph } e. On /\ Ord suc { (/) }) -> ( {z e. { (/) } | ph } e. suc { (/) } <-> suc {z e. { (/) } | ph } C_ suc { (/) }))
29	2, 27, 28	mp2an 404	. . . . 5 |- ( {z e. { (/) } | ph } e. suc { (/) } <-> suc {z e. { (/) } | ph } C_ suc { (/) })
30	25, 29	mpbir 134	. . . 4 |- {z e. { (/) } | ph } e. suc { (/) }
31		elsucg 4090	. . . . 5 |- ( {z e. { (/) } | ph } e. On -> ( {z e. { (/) } | ph } e. suc { (/) } <-> ( {z e. { (/) } | ph } e. { (/) } \/ {z e. { (/) } | ph } = { (/) })))
32	2, 31	ax-mp 7	. . . 4 |- ( {z e. { (/) } | ph } e. suc { (/) } <-> ( {z e. { (/) } | ph } e. { (/) } \/ {z e. { (/) } | ph } = { (/) }))
33	30, 32	mpbi 133	. . 3 |- ( {z e. { (/) } | ph } e. { (/) } \/ {z e. { (/) } | ph } = { (/) })
34		elsni 3374	. . . . 5 |- ( {z e. { (/) } | ph } e. { (/) } -> {z e. { (/) } | ph } = (/))
35		ordtriexmidlem2 4193	. . . . 5 |- ( {z e. { (/) } | ph } = (/) -> -. ph)
36	34, 35	syl 14	. . . 4 |- ( {z e. { (/) } | ph } e. { (/) } -> -. ph)
37		0ex 3858	. . . . 5 |- (/) e. _V
38		biidd 161	. . . . 5 |- (z = (/) -> (ph <-> ph))
39	37, 38	rabsnt 3419	. . . 4 |- ( {z e. { (/) } | ph } = { (/) } -> ph)
40	36, 39	orim12i 663	. . 3 |- (( {z e. { (/) } | ph } e. { (/) } \/ {z e. { (/) } | ph } = { (/) }) -> (-. ph \/ ph))
41	33, 40	ax-mp 7	. 2 |- (-. ph \/ ph)
42		orcom 634	. 2 |- ((-. ph \/ ph) <-> (ph \/ -. ph))
43	41, 42	mpbi 133	1 |- (ph \/ -. ph)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 616   = wceq 1228   e. wcel 1374  A.wral 2284   {crab 2288   C_ wss 2894   (/)c0 3201   {csn 3350   Ord word 4048   Oncon0 4049   suc csuc 4051

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057

This theorem is referenced by: (None)