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Theorem onsucelsucexmid 4440

Description: The converse of onsucelsucr 4419 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 4419 does hold, as seen at nnsucelsuc 6380. (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 4438	. . . 4 $\/$
2		0elon 4309	. . . . . 6
3		onsucelsucexmidlem 4439	. . . . . 6 $\/$
4	2, 3	pm3.2i 270	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2200	. . . . . . 7
7		suceq 4319	. . . . . . . 8
8	7	eleq1d 2206	. . . . . . 7
9	6, 8	imbi12d 233	. . . . . 6
10		eleq2 2201	. . . . . . 7 $\/$ $\/$
11		suceq 4319	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2207	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 233	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2798	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 422	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4320	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4328	. . . . . 6
20		p0ex 4107	. . . . . . 7
21	20	prid2 3625	. . . . . 6
22	19, 21	eqeltri 2210	. . . . 5
23		eqeq1 2144	. . . . . . 7
24	23	orbi1d 780	. . . . . 6 $\/$ $\/$
25	24	elrab3 2836	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4050	. . . . . . 7
28		nsuceq0g 4335	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2307	. . . . . 6
31	29, 30	mpbi 144	. . . . 5
32		pm2.53 711	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 120	. . 3 $\/$
35	19	eqeq1i 2145	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2145	. . . . . . . 8
37	31, 36	mtbi 659	. . . . . . 7
38	20	elsn 3538	. . . . . . 7
39	37, 38	mtbir 660	. . . . . 6
40		eleq2 2201	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 663	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 120	. . . 4 $\/$ $\/$
43		olc 700	. . . . 5 $\/$
44		eqeq1 2144	. . . . . . . 8
45	44	orbi1d 780	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2836	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 133	. . . 4 $\/$
49	42, 48	nsyl 617	. . 3 $\/$
50	34, 49	orim12i 748	. 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 697

wceq 1331

wcel 1480

wne 2306

wral 2414

crab 2418

cvv 2681

c0 3358

csn 3522

cpr 3523

con0 4280

csuc 4282

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288

This theorem is referenced by: ordsucunielexmid 4441

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