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Theorem onsucelsucexmid 4621

Description: The converse of onsucelsucr 4599 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 4599 does hold, as seen at nnsucelsuc 6635. (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 4619	. . . 4 $\/$
2		0elon 4482	. . . . . 6
3		onsucelsucexmidlem 4620	. . . . . 6 $\/$
4	2, 3	pm3.2i 272	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2292	. . . . . . 7
7		suceq 4492	. . . . . . . 8
8	7	eleq1d 2298	. . . . . . 7
9	6, 8	imbi12d 234	. . . . . 6
10		eleq2 2293	. . . . . . 7 $\/$ $\/$
11		suceq 4492	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2299	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 234	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2921	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 426	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4493	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4501	. . . . . 6
20		p0ex 4271	. . . . . . 7
21	20	prid2 3773	. . . . . 6
22	19, 21	eqeltri 2302	. . . . 5
23		eqeq1 2236	. . . . . . 7
24	23	orbi1d 796	. . . . . 6 $\/$ $\/$
25	24	elrab3 2960	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4210	. . . . . . 7
28		nsuceq0g 4508	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2401	. . . . . 6
31	29, 30	mpbi 145	. . . . 5
32		pm2.53 727	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 121	. . 3 $\/$
35	19	eqeq1i 2237	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2237	. . . . . . . 8
37	31, 36	mtbi 674	. . . . . . 7
38	20	elsn 3682	. . . . . . 7
39	37, 38	mtbir 675	. . . . . 6
40		eleq2 2293	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 678	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 121	. . . 4 $\/$ $\/$
43		olc 716	. . . . 5 $\/$
44		eqeq1 2236	. . . . . . . 8
45	44	orbi1d 796	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2960	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 134	. . . 4 $\/$
49	42, 48	nsyl 631	. . 3 $\/$
50	34, 49	orim12i 764	. 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 713

wceq 1395

wcel 2200

wne 2400

wral 2508

crab 2512

cvv 2799

c0 3491

csn 3666

cpr 3667

con0 4453

csuc 4455

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4201 ax-nul 4209 ax-pow 4257

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-uni 3888 df-tr 4182 df-iord 4456 df-on 4458 df-suc 4461

This theorem is referenced by: ordsucunielexmid 4622

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