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Theorem onsucelsucexmid 4672

Description: The converse of onsucelsucr 4650 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 4650 does hold, as seen at nnsucelsuc 6754. (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 4670	. . . 4 $\/$
2		0elon 4532	. . . . . 6
3		onsucelsucexmidlem 4671	. . . . . 6 $\/$
4	2, 3	pm3.2i 272	. . . . 5 $/\$ $\/$
5		onsucelsucexmid.1	. . . . 5
6		eleq1 2301	. . . . . . 7
7		suceq 4542	. . . . . . . 8
8	7	eleq1d 2307	. . . . . . 7
9	6, 8	imbi12d 234	. . . . . 6
10		eleq2 2302	. . . . . . 7 $\/$ $\/$
11		suceq 4542	. . . . . . . 8 $\/$ $\/$
12	11	eleq2d 2308	. . . . . . 7 $\/$ $\/$
13	10, 12	imbi12d 234	. . . . . 6 $\/$ $\/$ $\/$
14	9, 13	rspc2va 2944	. . . . 5 $/\$ $\/$ $/\$ $\/$ $\/$
15	4, 5, 14	mp2an 430	. . . 4 $\/$ $\/$
16	1, 15	ax-mp 5	. . 3 $\/$
17		elsuci 4543	. . 3 $\/$ $\/$ $\/$ $\/$
18	16, 17	ax-mp 5	. 2 $\/$ $\/$ $\/$
19		suc0 4551	. . . . . 6
20		p0ex 4320	. . . . . . 7
21	20	prid2 3814	. . . . . 6
22	19, 21	eqeltri 2311	. . . . 5
23		eqeq1 2245	. . . . . . 7
24	23	orbi1d 803	. . . . . 6 $\/$ $\/$
25	24	elrab3 2983	. . . . 5 $\/$ $\/$
26	22, 25	ax-mp 5	. . . 4 $\/$ $\/$
27		0ex 4255	. . . . . . 7
28		nsuceq0g 4558	. . . . . . 7
29	27, 28	ax-mp 5	. . . . . 6
30		df-ne 2421	. . . . . 6
31	29, 30	mpbi 145	. . . . 5
32		pm2.53 734	. . . . 5 $\/$
33	31, 32	mpi 15	. . . 4 $\/$
34	26, 33	sylbi 121	. . 3 $\/$
35	19	eqeq1i 2246	. . . . 5 $\/$ $\/$
36	19	eqeq1i 2246	. . . . . . . 8
37	31, 36	mtbi 681	. . . . . . 7
38	20	elsn 3721	. . . . . . 7
39	37, 38	mtbir 682	. . . . . 6
40		eleq2 2302	. . . . . 6 $\/$ $\/$
41	39, 40	mtbii 685	. . . . 5 $\/$ $\/$
42	35, 41	sylbi 121	. . . 4 $\/$ $\/$
43		olc 723	. . . . 5 $\/$
44		eqeq1 2245	. . . . . . . 8
45	44	orbi1d 803	. . . . . . 7 $\/$ $\/$
46	45	elrab3 2983	. . . . . 6 $\/$ $\/$
47	21, 46	ax-mp 5	. . . . 5 $\/$ $\/$
48	43, 47	sylibr 134	. . . 4 $\/$
49	42, 48	nsyl 637	. . 3 $\/$
50	34, 49	orim12i 771	. 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 720

wceq 1402

wcel 2209

wne 2420

wral 2528

crab 2532

cvv 2821

c0 3520

csn 3705

cpr 3706

con0 4503

csuc 4505

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-nul 4254 ax-pow 4306

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3687 df-sn 3711 df-pr 3712 df-uni 3931 df-tr 4225 df-iord 4506 df-on 4508 df-suc 4511

This theorem is referenced by: ordsucunielexmid 4673

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