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Theorem onsucelsucexmid 4255

Description: The converse of onsucelsucr 4234 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 4234 does hold, as seen at nnsucelsuc 6070. (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 4253	. . . 4 $\/$
2		0elon 4129	. . . . . 6
3		onsucelsucexmidlem 4254	. . . . . 6 $\/$
4	2, 3	pm3.2i 257	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2100	. . . . . . 7
7		suceq 4139	. . . . . . . 8
8	7	eleq1d 2106	. . . . . . 7
9	6, 8	imbi12d 223	. . . . . 6
10		eleq2 2101	. . . . . . 7 $\/$ $\/$
11		suceq 4139	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2107	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 223	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2663	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 402	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 7	. . 3 $\/$
17		elsuci 4140	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 7	. 2 $\/$ $\/$ $\/$
19		suc0 4148	. . . . . 6
20		p0ex 3939	. . . . . . 7
21	20	prid2 3477	. . . . . 6
22	19, 21	eqeltri 2110	. . . . 5
23		eqeq1 2046	. . . . . . 7
24	23	orbi1d 705	. . . . . 6 $\/$ $\/$
25	24	elrab3 2699	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 7	. . . 4 $\/$ $\/$
27		0ex 3884	. . . . . . 7
28		nsuceq0g 4155	. . . . . . 7
29	27, 28	ax-mp 7	. . . . . 6
30		df-ne 2206	. . . . . 6
31	29, 30	mpbi 133	. . . . 5
32		pm2.53 641	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 114	. . 3 $\/$
35	19	eqeq1i 2047	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2047	. . . . . . . 8
37	31, 36	mtbi 595	. . . . . . 7
38	20	elsn 3391	. . . . . . 7
39	37, 38	mtbir 596	. . . . . 6
40		eleq2 2101	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 599	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 114	. . . 4 $\/$ $\/$
43		olc 632	. . . . 5 $\/$
44		eqeq1 2046	. . . . . . . 8
45	44	orbi1d 705	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2699	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 7	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 137	. . . 4 $\/$
49	42, 48	nsyl 558	. . 3 $\/$
50	34, 49	orim12i 676	. 2 $\/$ $\/$ $\/$ $\/$
51	18, 50	ax-mp 7	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629

wceq 1243

wcel 1393

wne 2204

wral 2306

crab 2310

cvv 2557

c0 3224

csn 3375

cpr 3376

con0 4100

csuc 4102

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 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108

This theorem is referenced by: ordsucunielexmid 4256

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