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Theorem onsucelsucexmid 4383

Description: The converse of onsucelsucr 4362 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 4362 does hold, as seen at nnsucelsuc 6317. (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 4381	. . . 4 $\/$
2		0elon 4252	. . . . . 6
3		onsucelsucexmidlem 4382	. . . . . 6 $\/$
4	2, 3	pm3.2i 268	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2162	. . . . . . 7
7		suceq 4262	. . . . . . . 8
8	7	eleq1d 2168	. . . . . . 7
9	6, 8	imbi12d 233	. . . . . 6
10		eleq2 2163	. . . . . . 7 $\/$ $\/$
11		suceq 4262	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2169	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 233	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2757	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 420	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 7	. . 3 $\/$
17		elsuci 4263	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 7	. 2 $\/$ $\/$ $\/$
19		suc0 4271	. . . . . 6
20		p0ex 4052	. . . . . . 7
21	20	prid2 3577	. . . . . 6
22	19, 21	eqeltri 2172	. . . . 5
23		eqeq1 2106	. . . . . . 7
24	23	orbi1d 746	. . . . . 6 $\/$ $\/$
25	24	elrab3 2794	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 7	. . . 4 $\/$ $\/$
27		0ex 3995	. . . . . . 7
28		nsuceq0g 4278	. . . . . . 7
29	27, 28	ax-mp 7	. . . . . 6
30		df-ne 2268	. . . . . 6
31	29, 30	mpbi 144	. . . . 5
32		pm2.53 682	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 120	. . 3 $\/$
35	19	eqeq1i 2107	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2107	. . . . . . . 8
37	31, 36	mtbi 636	. . . . . . 7
38	20	elsn 3490	. . . . . . 7
39	37, 38	mtbir 637	. . . . . 6
40		eleq2 2163	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 640	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 120	. . . 4 $\/$ $\/$
43		olc 673	. . . . 5 $\/$
44		eqeq1 2106	. . . . . . . 8
45	44	orbi1d 746	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2794	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 7	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 133	. . . 4 $\/$
49	42, 48	nsyl 598	. . 3 $\/$
50	34, 49	orim12i 717	. 2 $\/$ $\/$ $\/$ $\/$
51	18, 50	ax-mp 7	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 670

wceq 1299

wcel 1448

wne 2267

wral 2375

crab 2379

cvv 2641

c0 3310

csn 3474

cpr 3475

con0 4223

csuc 4225

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038

This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-uni 3684 df-tr 3967 df-iord 4226 df-on 4228 df-suc 4231

This theorem is referenced by: ordsucunielexmid 4384

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