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Theorem onsucelsucexmid 4567

Description: The converse of onsucelsucr 4545 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 4545 does hold, as seen at nnsucelsuc 6558. (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 4565	. . . 4 $\/$
2		0elon 4428	. . . . . 6
3		onsucelsucexmidlem 4566	. . . . . 6 $\/$
4	2, 3	pm3.2i 272	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2259	. . . . . . 7
7		suceq 4438	. . . . . . . 8
8	7	eleq1d 2265	. . . . . . 7
9	6, 8	imbi12d 234	. . . . . 6
10		eleq2 2260	. . . . . . 7 $\/$ $\/$
11		suceq 4438	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2266	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 234	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2882	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 426	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4439	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4447	. . . . . 6
20		p0ex 4222	. . . . . . 7
21	20	prid2 3730	. . . . . 6
22	19, 21	eqeltri 2269	. . . . 5
23		eqeq1 2203	. . . . . . 7
24	23	orbi1d 792	. . . . . 6 $\/$ $\/$
25	24	elrab3 2921	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4161	. . . . . . 7
28		nsuceq0g 4454	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2368	. . . . . 6
31	29, 30	mpbi 145	. . . . 5
32		pm2.53 723	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 121	. . 3 $\/$
35	19	eqeq1i 2204	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2204	. . . . . . . 8
37	31, 36	mtbi 671	. . . . . . 7
38	20	elsn 3639	. . . . . . 7
39	37, 38	mtbir 672	. . . . . 6
40		eleq2 2260	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 675	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 121	. . . 4 $\/$ $\/$
43		olc 712	. . . . 5 $\/$
44		eqeq1 2203	. . . . . . . 8
45	44	orbi1d 792	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2921	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 134	. . . 4 $\/$
49	42, 48	nsyl 629	. . 3 $\/$
50	34, 49	orim12i 760	. 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 709

wceq 1364

wcel 2167

wne 2367

wral 2475

crab 2479

cvv 2763

c0 3451

csn 3623

cpr 3624

con0 4399

csuc 4401

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-nul 4160 ax-pow 4208

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-uni 3841 df-tr 4133 df-iord 4402 df-on 4404 df-suc 4407

This theorem is referenced by: ordsucunielexmid 4568

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