nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4725

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4639 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6710 or nntri3or 6704), 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 3313	. . . 4 $\/$ $/\$
2		peano1 4698	. . . . 5
3		suc0 4514	. . . . . 6
4		peano2 4699	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2305	. . . . 5
7		prssi 3836	. . . . 5 $/\$
8	2, 6, 7	mp2an 426	. . . 4
9	1, 8	sstri 3237	. . 3 $\/$ $/\$
10		eqid 2231	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4636	. . 3 $\/$ $/\$
12		pp0ex 4285	. . . . 5
13	12	rabex 4239	. . . 4 $\/$ $/\$
14		sseq1 3251	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2295	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1873	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 473	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2295	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 673	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 230	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1872	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 473	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1873	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 234	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2859	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 426	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4637	. 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 2202

crab 2515

wss 3201

c0 3496

csn 3673

cpr 3674

csuc 4468

com 4694

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 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-uni 3899 df-int 3934 df-suc 4474 df-iom 4695

This theorem is referenced by: (None)

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