nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4743

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4657 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6732 or nntri3or 6726), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)

**Hypothesis**
Ref	Expression
nnregexmid.1	$/\$ $/\$

**Assertion**
Ref	Expression
nnregexmid	$\/$

Distinct variable group:

Proof of Theorem nnregexmid Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3323	. . . 4 $\/$ $/\$
2		peano1 4716	. . . . 5
3		suc0 4532	. . . . . 6
4		peano2 4717	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2306	. . . . 5
7		prssi 3852	. . . . 5 $/\$
8	2, 6, 7	mp2an 426	. . . 4
9	1, 8	sstri 3247	. . 3 $\/$ $/\$
10		eqid 2232	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4654	. . 3 $\/$ $/\$
12		pp0ex 4302	. . . . 5
13	12	rabex 4256	. . . 4 $\/$ $/\$
14		sseq1 3261	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2296	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1874	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 473	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2296	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 673	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 230	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1873	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 473	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1874	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 234	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2869	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 426	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4655	. 2 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$
29	27, 28	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 716

wal 1396

wceq 1398

wex 1541

wcel 2203

crab 2524

wss 3211

c0 3508

csn 3689

cpr 3690

csuc 4486

com 4712

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2815 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-pw 3671 df-sn 3695 df-pr 3696 df-uni 3915 df-int 3950 df-suc 4492 df-iom 4713

This theorem is referenced by: (None)

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