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Theorem onsucelsucexmid 4628

Description: The converse of onsucelsucr 4606 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 4606 does hold, as seen at nnsucelsuc 6658. (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 4626	. . . 4 $\/$
2		0elon 4489	. . . . . 6
3		onsucelsucexmidlem 4627	. . . . . 6 $\/$
4	2, 3	pm3.2i 272	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2294	. . . . . . 7
7		suceq 4499	. . . . . . . 8
8	7	eleq1d 2300	. . . . . . 7
9	6, 8	imbi12d 234	. . . . . 6
10		eleq2 2295	. . . . . . 7 $\/$ $\/$
11		suceq 4499	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2301	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 234	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2924	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 426	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4500	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4508	. . . . . 6
20		p0ex 4278	. . . . . . 7
21	20	prid2 3778	. . . . . 6
22	19, 21	eqeltri 2304	. . . . 5
23		eqeq1 2238	. . . . . . 7
24	23	orbi1d 798	. . . . . 6 $\/$ $\/$
25	24	elrab3 2963	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4216	. . . . . . 7
28		nsuceq0g 4515	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2403	. . . . . 6
31	29, 30	mpbi 145	. . . . 5
32		pm2.53 729	. . . . 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 676	. . . . . . 7
38	20	elsn 3685	. . . . . . 7
39	37, 38	mtbir 677	. . . . . 6
40		eleq2 2295	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 680	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 121	. . . 4 $\/$ $\/$
43		olc 718	. . . . 5 $\/$
44		eqeq1 2238	. . . . . . . 8
45	44	orbi1d 798	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2963	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 134	. . . 4 $\/$
49	42, 48	nsyl 633	. . 3 $\/$
50	34, 49	orim12i 766	. 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 715

wceq 1397

wcel 2202

wne 2402

wral 2510

crab 2514

cvv 2802

c0 3494

csn 3669

cpr 3670

con0 4460

csuc 4462

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-nul 4215 ax-pow 4264

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-uni 3894 df-tr 4188 df-iord 4463 df-on 4465 df-suc 4468

This theorem is referenced by: ordsucunielexmid 4629

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