lzenom - Mathbox for Stefan O'Rear

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Theorem lzenom 37333

Description: Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)

**Assertion**
Ref	Expression
lzenom	$\$

Proof of Theorem lzenom Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zex 11386	. . . 4
2		difexg 4808	. . . 4 $\$
3	1, 2	mp1i 13	. . 3 $\$
4		nnex 11026	. . . 4
5	4	a1i 11	. . 3
6		ovex 6678	. . . 4
7	6	2a1i 12	. . 3 $\$
8		ovex 6678	. . . 4
9	8	2a1i 12	. . 3
10		simpl 473	. . . . . . . . . 10 $/\$ $/\$
11	10	peano2zd 11485	. . . . . . . . 9 $/\$ $/\$
12		simprl 794	. . . . . . . . 9 $/\$ $/\$
13	11, 12	zsubcld 11487	. . . . . . . 8 $/\$ $/\$
14		zre 11381	. . . . . . . . . 10
15	14	ad2antrl 764	. . . . . . . . 9 $/\$ $/\$
16	11	zred 11482	. . . . . . . . 9 $/\$ $/\$
17		1red 10055	. . . . . . . . 9 $/\$ $/\$
18		simprr 796	. . . . . . . . . 10 $/\$ $/\$
19		zcn 11382	. . . . . . . . . . . 12
20	19	adantr 481	. . . . . . . . . . 11 $/\$ $/\$
21		ax-1cn 9994	. . . . . . . . . . 11
22		pncan 10287	. . . . . . . . . . 11 $/\$
23	20, 21, 22	sylancl 694	. . . . . . . . . 10 $/\$ $/\$
24	18, 23	breqtrrd 4681	. . . . . . . . 9 $/\$ $/\$
25	15, 16, 17, 24	lesubd 10631	. . . . . . . 8 $/\$ $/\$
26	11	zcnd 11483	. . . . . . . . . 10 $/\$ $/\$
27		zcn 11382	. . . . . . . . . . 11
28	27	ad2antrl 764	. . . . . . . . . 10 $/\$ $/\$
29	26, 28	nncand 10397	. . . . . . . . 9 $/\$ $/\$
30	29	eqcomd 2628	. . . . . . . 8 $/\$ $/\$
31	13, 25, 30	jca31 557	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	31	adantrr 753	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
33		eleq1 2689	. . . . . . . . 9
34		breq2 4657	. . . . . . . . 9
35	33, 34	anbi12d 747	. . . . . . . 8 $/\$ $/\$
36		oveq2 6658	. . . . . . . . 9
37	36	eqeq2d 2632	. . . . . . . 8
38	35, 37	anbi12d 747	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	38	ad2antll 765	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	32, 39	mpbird 247	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
41		simpl 473	. . . . . . . . . 10 $/\$ $/\$
42	41	peano2zd 11485	. . . . . . . . 9 $/\$ $/\$
43		simprl 794	. . . . . . . . 9 $/\$ $/\$
44	42, 43	zsubcld 11487	. . . . . . . 8 $/\$ $/\$
45	42	zred 11482	. . . . . . . . 9 $/\$ $/\$
46		zre 11381	. . . . . . . . . 10
47	46	adantr 481	. . . . . . . . 9 $/\$ $/\$
48		zre 11381	. . . . . . . . . 10
49	48	ad2antrl 764	. . . . . . . . 9 $/\$ $/\$
50	47	recnd 10068	. . . . . . . . . . 11 $/\$ $/\$
51		pncan2 10288	. . . . . . . . . . 11 $/\$
52	50, 21, 51	sylancl 694	. . . . . . . . . 10 $/\$ $/\$
53		simprr 796	. . . . . . . . . 10 $/\$ $/\$
54	52, 53	eqbrtrd 4675	. . . . . . . . 9 $/\$ $/\$
55	45, 47, 49, 54	subled 10630	. . . . . . . 8 $/\$ $/\$
56	42	zcnd 11483	. . . . . . . . . 10 $/\$ $/\$
57		zcn 11382	. . . . . . . . . . 11
58	57	ad2antrl 764	. . . . . . . . . 10 $/\$ $/\$
59	56, 58	nncand 10397	. . . . . . . . 9 $/\$ $/\$
60	59	eqcomd 2628	. . . . . . . 8 $/\$ $/\$
61	44, 55, 60	jca31 557	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
62	61	adantrr 753	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
63		eleq1 2689	. . . . . . . . 9
64		breq1 4656	. . . . . . . . 9
65	63, 64	anbi12d 747	. . . . . . . 8 $/\$ $/\$
66		oveq2 6658	. . . . . . . . 9
67	66	eqeq2d 2632	. . . . . . . 8
68	65, 67	anbi12d 747	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
69	68	ad2antll 765	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	62, 69	mpbird 247	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
71	40, 70	impbida 877	. . . 4 $/\$ $/\$ $/\$ $/\$
72		ellz1 37330	. . . . 5 $\$ $/\$
73	72	anbi1d 741	. . . 4 $\$ $/\$ $/\$ $/\$
74		elnnz1 11403	. . . . . 6 $/\$
75	74	a1i 11	. . . . 5 $/\$
76	75	anbi1d 741	. . . 4 $/\$ $/\$ $/\$
77	71, 73, 76	3bitr4d 300	. . 3 $\$ $/\$ $/\$
78	3, 5, 7, 9, 77	en2d 7991	. 2 $\$
79		nnenom 12779	. 2
80		entr 8008	. 2 $\$ $/\$ $\$
81	78, 79, 80	sylancl 694	1 $\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

cvv 3200 $\$ cdif 3571 class class class wbr 4653

cfv 5888 (class class class)co 6650

com 7065

cen 7952

cc 9934

cr 9935

c1 9937

caddc 9939

cle 10075

cmin 10266

cn 11020

cz 11377

cuz 11687

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688

This theorem is referenced by: diophin 37336 diophren 37377

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