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Theorem onsucelsucexmid 4507

Description: The converse of onsucelsucr 4485 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 4485 does hold, as seen at nnsucelsuc 6459. (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 4505	. . . 4 $\/$
2		0elon 4370	. . . . . 6
3		onsucelsucexmidlem 4506	. . . . . 6 $\/$
4	2, 3	pm3.2i 270	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2229	. . . . . . 7
7		suceq 4380	. . . . . . . 8
8	7	eleq1d 2235	. . . . . . 7
9	6, 8	imbi12d 233	. . . . . 6
10		eleq2 2230	. . . . . . 7 $\/$ $\/$
11		suceq 4380	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2236	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 233	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2844	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 423	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4381	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4389	. . . . . 6
20		p0ex 4167	. . . . . . 7
21	20	prid2 3683	. . . . . 6
22	19, 21	eqeltri 2239	. . . . 5
23		eqeq1 2172	. . . . . . 7
24	23	orbi1d 781	. . . . . 6 $\/$ $\/$
25	24	elrab3 2883	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4109	. . . . . . 7
28		nsuceq0g 4396	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2337	. . . . . 6
31	29, 30	mpbi 144	. . . . 5
32		pm2.53 712	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 120	. . 3 $\/$
35	19	eqeq1i 2173	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2173	. . . . . . . 8
37	31, 36	mtbi 660	. . . . . . 7
38	20	elsn 3592	. . . . . . 7
39	37, 38	mtbir 661	. . . . . 6
40		eleq2 2230	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 664	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 120	. . . 4 $\/$ $\/$
43		olc 701	. . . . 5 $\/$
44		eqeq1 2172	. . . . . . . 8
45	44	orbi1d 781	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2883	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 133	. . . 4 $\/$
49	42, 48	nsyl 618	. . 3 $\/$
50	34, 49	orim12i 749	. 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 698

wceq 1343

wcel 2136

wne 2336

wral 2444

crab 2448

cvv 2726

c0 3409

csn 3576

cpr 3577

con0 4341

csuc 4343

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-nul 4108 ax-pow 4153

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-uni 3790 df-tr 4081 df-iord 4344 df-on 4346 df-suc 4349

This theorem is referenced by: ordsucunielexmid 4508

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