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Theorem onsucelsucexmid 4576

Description: The converse of onsucelsucr 4554 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 4554 does hold, as seen at nnsucelsuc 6567. (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 4574	. . . 4 $\/$
2		0elon 4437	. . . . . 6
3		onsucelsucexmidlem 4575	. . . . . 6 $\/$
4	2, 3	pm3.2i 272	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2267	. . . . . . 7
7		suceq 4447	. . . . . . . 8
8	7	eleq1d 2273	. . . . . . 7
9	6, 8	imbi12d 234	. . . . . 6
10		eleq2 2268	. . . . . . 7 $\/$ $\/$
11		suceq 4447	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2274	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 234	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2890	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 426	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4448	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4456	. . . . . 6
20		p0ex 4231	. . . . . . 7
21	20	prid2 3739	. . . . . 6
22	19, 21	eqeltri 2277	. . . . 5
23		eqeq1 2211	. . . . . . 7
24	23	orbi1d 792	. . . . . 6 $\/$ $\/$
25	24	elrab3 2929	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4170	. . . . . . 7
28		nsuceq0g 4463	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2376	. . . . . 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 2212	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2212	. . . . . . . 8
37	31, 36	mtbi 671	. . . . . . 7
38	20	elsn 3648	. . . . . . 7
39	37, 38	mtbir 672	. . . . . 6
40		eleq2 2268	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 675	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 121	. . . 4 $\/$ $\/$
43		olc 712	. . . . 5 $\/$
44		eqeq1 2211	. . . . . . . 8
45	44	orbi1d 792	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2929	. . . . . 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 1372

wcel 2175

wne 2375

wral 2483

crab 2487

cvv 2771

c0 3459

csn 3632

cpr 3633

con0 4408

csuc 4410

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-nul 4169 ax-pow 4217

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-ral 2488 df-rex 2489 df-rab 2492 df-v 2773 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-pw 3617 df-sn 3638 df-pr 3639 df-uni 3850 df-tr 4142 df-iord 4411 df-on 4413 df-suc 4416

This theorem is referenced by: ordsucunielexmid 4577

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