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Theorem onsucelsucexmid 4586

Description: The converse of onsucelsucr 4564 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 4564 does hold, as seen at nnsucelsuc 6590. (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 4584	. . . 4 $\/$
2		0elon 4447	. . . . . 6
3		onsucelsucexmidlem 4585	. . . . . 6 $\/$
4	2, 3	pm3.2i 272	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2269	. . . . . . 7
7		suceq 4457	. . . . . . . 8
8	7	eleq1d 2275	. . . . . . 7
9	6, 8	imbi12d 234	. . . . . 6
10		eleq2 2270	. . . . . . 7 $\/$ $\/$
11		suceq 4457	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2276	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 234	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2895	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 426	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4458	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4466	. . . . . 6
20		p0ex 4240	. . . . . . 7
21	20	prid2 3745	. . . . . 6
22	19, 21	eqeltri 2279	. . . . 5
23		eqeq1 2213	. . . . . . 7
24	23	orbi1d 793	. . . . . 6 $\/$ $\/$
25	24	elrab3 2934	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4179	. . . . . . 7
28		nsuceq0g 4473	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2378	. . . . . 6
31	29, 30	mpbi 145	. . . . 5
32		pm2.53 724	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 121	. . 3 $\/$
35	19	eqeq1i 2214	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2214	. . . . . . . 8
37	31, 36	mtbi 672	. . . . . . 7
38	20	elsn 3654	. . . . . . 7
39	37, 38	mtbir 673	. . . . . 6
40		eleq2 2270	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 676	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 121	. . . 4 $\/$ $\/$
43		olc 713	. . . . 5 $\/$
44		eqeq1 2213	. . . . . . . 8
45	44	orbi1d 793	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2934	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 134	. . . 4 $\/$
49	42, 48	nsyl 629	. . 3 $\/$
50	34, 49	orim12i 761	. 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 710

wceq 1373

wcel 2177

wne 2377

wral 2485

crab 2489

cvv 2773

c0 3464

csn 3638

cpr 3639

con0 4418

csuc 4420

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-nul 4178 ax-pow 4226

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-ral 2490 df-rex 2491 df-rab 2494 df-v 2775 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-pw 3623 df-sn 3644 df-pr 3645 df-uni 3857 df-tr 4151 df-iord 4421 df-on 4423 df-suc 4426

This theorem is referenced by: ordsucunielexmid 4587

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