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Theorem onsucelsucexmid 4634

Description: The converse of onsucelsucr 4612 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 4612 does hold, as seen at nnsucelsuc 6702. (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 4632	. . . 4 $\/$
2		0elon 4495	. . . . . 6
3		onsucelsucexmidlem 4633	. . . . . 6 $\/$
4	2, 3	pm3.2i 272	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2294	. . . . . . 7
7		suceq 4505	. . . . . . . 8
8	7	eleq1d 2300	. . . . . . 7
9	6, 8	imbi12d 234	. . . . . 6
10		eleq2 2295	. . . . . . 7 $\/$ $\/$
11		suceq 4505	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2301	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 234	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2925	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 426	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4506	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4514	. . . . . 6
20		p0ex 4284	. . . . . . 7
21	20	prid2 3782	. . . . . 6
22	19, 21	eqeltri 2304	. . . . 5
23		eqeq1 2238	. . . . . . 7
24	23	orbi1d 799	. . . . . 6 $\/$ $\/$
25	24	elrab3 2964	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4221	. . . . . . 7
28		nsuceq0g 4521	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2404	. . . . . 6
31	29, 30	mpbi 145	. . . . 5
32		pm2.53 730	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 121	. . 3 $\/$
35	19	eqeq1i 2239	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2239	. . . . . . . 8
37	31, 36	mtbi 677	. . . . . . 7
38	20	elsn 3689	. . . . . . 7
39	37, 38	mtbir 678	. . . . . 6
40		eleq2 2295	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 681	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 121	. . . 4 $\/$ $\/$
43		olc 719	. . . . 5 $\/$
44		eqeq1 2238	. . . . . . . 8
45	44	orbi1d 799	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2964	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 134	. . . 4 $\/$
49	42, 48	nsyl 633	. . 3 $\/$
50	34, 49	orim12i 767	. 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 716

wceq 1398

wcel 2202

wne 2403

wral 2511

crab 2515

cvv 2803

c0 3496

csn 3673

cpr 3674

con0 4466

csuc 4468

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-nul 4220 ax-pow 4270

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-uni 3899 df-tr 4193 df-iord 4469 df-on 4471 df-suc 4474

This theorem is referenced by: ordsucunielexmid 4635

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