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Theorem onsucelsucexmid 4273

Description: The converse of onsucelsucr 4252 implies excluded middle. On the other hand, if

is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4252 does hold, as seen at nnsucelsuc 6093. (Contributed by Jim Kingdon, 2-Aug-2019.)

**Hypothesis**
Ref	Expression
onsucelsucexmid.1

**Assertion**
Ref	Expression
onsucelsucexmid	$\/$

Distinct variable group:

Proof of Theorem onsucelsucexmid Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onsucelsucexmidlem1 4271	. . . 4 $\/$
2		0elon 4147	. . . . . 6
3		onsucelsucexmidlem 4272	. . . . . 6 $\/$
4	2, 3	pm3.2i 266	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2141	. . . . . . 7
7		suceq 4157	. . . . . . . 8
8	7	eleq1d 2147	. . . . . . 7
9	6, 8	imbi12d 232	. . . . . 6
10		eleq2 2142	. . . . . . 7 $\/$ $\/$
11		suceq 4157	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2148	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 232	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2714	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 416	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 7	. . 3 $\/$
17		elsuci 4158	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 7	. 2 $\/$ $\/$ $\/$
19		suc0 4166	. . . . . 6
20		p0ex 3959	. . . . . . 7
21	20	prid2 3499	. . . . . 6
22	19, 21	eqeltri 2151	. . . . 5
23		eqeq1 2087	. . . . . . 7
24	23	orbi1d 737	. . . . . 6 $\/$ $\/$
25	24	elrab3 2750	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 7	. . . 4 $\/$ $\/$
27		0ex 3905	. . . . . . 7
28		nsuceq0g 4173	. . . . . . 7
29	27, 28	ax-mp 7	. . . . . 6
30		df-ne 2246	. . . . . 6
31	29, 30	mpbi 143	. . . . 5
32		pm2.53 673	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 119	. . 3 $\/$
35	19	eqeq1i 2088	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2088	. . . . . . . 8
37	31, 36	mtbi 627	. . . . . . 7
38	20	elsn 3414	. . . . . . 7
39	37, 38	mtbir 628	. . . . . 6
40		eleq2 2142	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 631	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 119	. . . 4 $\/$ $\/$
43		olc 664	. . . . 5 $\/$
44		eqeq1 2087	. . . . . . . 8
45	44	orbi1d 737	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2750	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 7	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 132	. . . 4 $\/$
49	42, 48	nsyl 590	. . 3 $\/$
50	34, 49	orim12i 708	. 2 $\/$ $\/$ $\/$ $\/$
51	18, 50	ax-mp 7	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102

wb 103 $\/$ wo 661

wceq 1284

wcel 1433

wne 2245

wral 2348

crab 2352

cvv 2601

c0 3251

csn 3398

cpr 3399

con0 4118

csuc 4120

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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126

This theorem is referenced by: ordsucunielexmid 4274

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