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Theorem onsucelsucexmid 4523

Description: The converse of onsucelsucr 4501 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 4501 does hold, as seen at nnsucelsuc 6482. (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 4521	. . . 4 $\/$
2		0elon 4386	. . . . . 6
3		onsucelsucexmidlem 4522	. . . . . 6 $\/$
4	2, 3	pm3.2i 272	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2238	. . . . . . 7
7		suceq 4396	. . . . . . . 8
8	7	eleq1d 2244	. . . . . . 7
9	6, 8	imbi12d 234	. . . . . 6
10		eleq2 2239	. . . . . . 7 $\/$ $\/$
11		suceq 4396	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2245	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 234	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2853	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 426	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4397	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4405	. . . . . 6
20		p0ex 4183	. . . . . . 7
21	20	prid2 3696	. . . . . 6
22	19, 21	eqeltri 2248	. . . . 5
23		eqeq1 2182	. . . . . . 7
24	23	orbi1d 791	. . . . . 6 $\/$ $\/$
25	24	elrab3 2892	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4125	. . . . . . 7
28		nsuceq0g 4412	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2346	. . . . . 6
31	29, 30	mpbi 145	. . . . 5
32		pm2.53 722	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 121	. . 3 $\/$
35	19	eqeq1i 2183	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2183	. . . . . . . 8
37	31, 36	mtbi 670	. . . . . . 7
38	20	elsn 3605	. . . . . . 7
39	37, 38	mtbir 671	. . . . . 6
40		eleq2 2239	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 674	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 121	. . . 4 $\/$ $\/$
43		olc 711	. . . . 5 $\/$
44		eqeq1 2182	. . . . . . . 8
45	44	orbi1d 791	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2892	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 134	. . . 4 $\/$
49	42, 48	nsyl 628	. . 3 $\/$
50	34, 49	orim12i 759	. 2 $\/$ $\/$ $\/$ $\/$
51	18, 50	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708

wceq 1353

wcel 2146

wne 2345

wral 2453

crab 2457

cvv 2735

c0 3420

csn 3589

cpr 3590

con0 4357

csuc 4359

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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-nul 4124 ax-pow 4169

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-uni 3806 df-tr 4097 df-iord 4360 df-on 4362 df-suc 4365

This theorem is referenced by: ordsucunielexmid 4524

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