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Theorem onsucelsucexmid 4531

Description: The converse of onsucelsucr 4509 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 4509 does hold, as seen at nnsucelsuc 6494. (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 4529	. . . 4 $\/$
2		0elon 4394	. . . . . 6
3		onsucelsucexmidlem 4530	. . . . . 6 $\/$
4	2, 3	pm3.2i 272	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2240	. . . . . . 7
7		suceq 4404	. . . . . . . 8
8	7	eleq1d 2246	. . . . . . 7
9	6, 8	imbi12d 234	. . . . . 6
10		eleq2 2241	. . . . . . 7 $\/$ $\/$
11		suceq 4404	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2247	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 234	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2857	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 426	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4405	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4413	. . . . . 6
20		p0ex 4190	. . . . . . 7
21	20	prid2 3701	. . . . . 6
22	19, 21	eqeltri 2250	. . . . 5
23		eqeq1 2184	. . . . . . 7
24	23	orbi1d 791	. . . . . 6 $\/$ $\/$
25	24	elrab3 2896	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4132	. . . . . . 7
28		nsuceq0g 4420	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2348	. . . . . 6
31	29, 30	mpbi 145	. . . . 5
32		pm2.53 722	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 121	. . 3 $\/$
35	19	eqeq1i 2185	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2185	. . . . . . . 8
37	31, 36	mtbi 670	. . . . . . 7
38	20	elsn 3610	. . . . . . 7
39	37, 38	mtbir 671	. . . . . 6
40		eleq2 2241	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 674	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 121	. . . 4 $\/$ $\/$
43		olc 711	. . . . 5 $\/$
44		eqeq1 2184	. . . . . . . 8
45	44	orbi1d 791	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2896	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 134	. . . 4 $\/$
49	42, 48	nsyl 628	. . 3 $\/$
50	34, 49	orim12i 759	. 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 708

wceq 1353

wcel 2148

wne 2347

wral 2455

crab 2459

cvv 2739

c0 3424

csn 3594

cpr 3595

con0 4365

csuc 4367

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-uni 3812 df-tr 4104 df-iord 4368 df-on 4370 df-suc 4373

This theorem is referenced by: ordsucunielexmid 4532

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