nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4605

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4519 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6478 or nntri3or 6472), 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 3232	. . . 4 $\/$ $/\$
2		peano1 4578	. . . . 5
3		suc0 4396	. . . . . 6
4		peano2 4579	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2244	. . . . 5
7		prssi 3738	. . . . 5 $/\$
8	2, 6, 7	mp2an 424	. . . 4
9	1, 8	sstri 3156	. . 3 $\/$ $/\$
10		eqid 2170	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4516	. . 3 $\/$ $/\$
12		pp0ex 4175	. . . . 5
13	12	rabex 4133	. . . 4 $\/$ $/\$
14		sseq1 3170	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2234	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1818	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 470	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2234	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 662	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 229	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1817	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 470	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1818	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 233	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2784	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 424	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4517	. 2 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$
29	27, 28	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 703

wal 1346

wceq 1348

wex 1485

wcel 2141

crab 2452

wss 3121

c0 3414

csn 3583

cpr 3584

csuc 4350

com 4574

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-suc 4356 df-iom 4575

This theorem is referenced by: (None)

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