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Theorem onsucelsucexmid 4514

Description: The converse of onsucelsucr 4492 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 4492 does hold, as seen at nnsucelsuc 6470. (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 4512	. . . 4 $\/$
2		0elon 4377	. . . . . 6
3		onsucelsucexmidlem 4513	. . . . . 6 $\/$
4	2, 3	pm3.2i 270	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2233	. . . . . . 7
7		suceq 4387	. . . . . . . 8
8	7	eleq1d 2239	. . . . . . 7
9	6, 8	imbi12d 233	. . . . . 6
10		eleq2 2234	. . . . . . 7 $\/$ $\/$
11		suceq 4387	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2240	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 233	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2848	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 424	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4388	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4396	. . . . . 6
20		p0ex 4174	. . . . . . 7
21	20	prid2 3690	. . . . . 6
22	19, 21	eqeltri 2243	. . . . 5
23		eqeq1 2177	. . . . . . 7
24	23	orbi1d 786	. . . . . 6 $\/$ $\/$
25	24	elrab3 2887	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4116	. . . . . . 7
28		nsuceq0g 4403	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2341	. . . . . 6
31	29, 30	mpbi 144	. . . . 5
32		pm2.53 717	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 120	. . 3 $\/$
35	19	eqeq1i 2178	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2178	. . . . . . . 8
37	31, 36	mtbi 665	. . . . . . 7
38	20	elsn 3599	. . . . . . 7
39	37, 38	mtbir 666	. . . . . 6
40		eleq2 2234	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 669	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 120	. . . 4 $\/$ $\/$
43		olc 706	. . . . 5 $\/$
44		eqeq1 2177	. . . . . . . 8
45	44	orbi1d 786	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2887	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 133	. . . 4 $\/$
49	42, 48	nsyl 623	. . 3 $\/$
50	34, 49	orim12i 754	. 2 $\/$ $\/$ $\/$ $\/$
51	18, 50	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703

wceq 1348

wcel 2141

wne 2340

wral 2448

crab 2452

cvv 2730

c0 3414

csn 3583

cpr 3584

con0 4348

csuc 4350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356

This theorem is referenced by: ordsucunielexmid 4515

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