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Theorem mreexexdOLD 16309

Description: Obsolete proof of mreexexd 16308 as of 2-Jun-2021. Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if

and

are disjoint from

is independent,

is contained in the closure of

, and either

is finite, then there is a subset

equinumerous to

such that

is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either

$\$

is finite. The theorem is proven by induction using mreexexlem3d 16306 for the base case and mreexexlem4d 16307 for the induction step. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
mreexexlem2d.1	Moore
mreexexlem2d.2	mrCls
mreexexlem2d.3	mrInd
mreexexlem2d.4	$\$
mreexexlem2d.5	$\$
mreexexlem2d.6	$\$
mreexexlem2d.7
mreexexlem2d.8
mreexexd.9	$\/$

**Assertion**
Ref	Expression
mreexexdOLD	$/\$

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

(

)

(

)

(

)

(

)

Proof of Theorem mreexexdOLD Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mreexexlem2d.1	. . 3 Moore
2	1	elfvexd 6222	. 2
3		mreexexlem2d.5	. 2 $\$
4		mreexexlem2d.6	. 2 $\$
5		mreexexlem2d.7	. 2
6		mreexexlem2d.8	. 2
7		cardon 8770	. . . . 5
8	7	onordi 5832	. . . 4
9		cardon 8770	. . . . 5
10	9	onordi 5832	. . . 4
11		ordtri2or3 5824	. . . 4 $/\$ $\/$
12	8, 10, 11	mp2an 708	. . 3 $\/$
13	3	difss2d 3740	. . . . . . . 8
14	2, 13	ssexd 4805	. . . . . . 7
15	14	cardidd 9371	. . . . . 6
16	15	ensymd 8007	. . . . 5
17		breq2 4657	. . . . 5
18	16, 17	syl5ibcom 235	. . . 4
19	4	difss2d 3740	. . . . . . . 8
20	2, 19	ssexd 4805	. . . . . . 7
21	20	cardidd 9371	. . . . . 6
22	21	ensymd 8007	. . . . 5
23		breq2 4657	. . . . 5
24	22, 23	syl5ibcom 235	. . . 4
25	18, 24	orim12d 883	. . 3 $\/$ $\/$
26	12, 25	mpi 20	. 2 $\/$
27		mreexexd.9	. . . . 5 $\/$
28		ficardom 8787	. . . . . 6
29		ficardom 8787	. . . . . 6
30	28, 29	orim12i 538	. . . . 5 $\/$ $\/$
31	27, 30	syl 17	. . . 4 $\/$
32		ordom 7074	. . . . 5
33		ordelinel 5825	. . . . 5 $/\$ $/\$ $\/$
34	8, 10, 32, 33	mp3an 1424	. . . 4 $\/$
35	31, 34	sylibr 224	. . 3
36		breq2 4657	. . . . . . . . . 10
37		breq2 4657	. . . . . . . . . 10
38	36, 37	orbi12d 746	. . . . . . . . 9 $\/$ $\/$
39	38	3anbi1d 1403	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$
40	39	imbi1d 331	. . . . . . 7 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$
41	40	2ralbidv 2989	. . . . . 6 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
42	41	albidv 1849	. . . . 5 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
43	42	imbi2d 330	. . . 4 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
44		breq2 4657	. . . . . . . . . 10
45		breq2 4657	. . . . . . . . . 10
46	44, 45	orbi12d 746	. . . . . . . . 9 $\/$ $\/$
47	46	3anbi1d 1403	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$
48	47	imbi1d 331	. . . . . . 7 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$
49	48	2ralbidv 2989	. . . . . 6 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
50	49	albidv 1849	. . . . 5 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
51	50	imbi2d 330	. . . 4 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
52		breq2 4657	. . . . . . . . . 10
53		breq2 4657	. . . . . . . . . 10
54	52, 53	orbi12d 746	. . . . . . . . 9 $\/$ $\/$
55	54	3anbi1d 1403	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$
56	55	imbi1d 331	. . . . . . 7 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$
57	56	2ralbidv 2989	. . . . . 6 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
58	57	albidv 1849	. . . . 5 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
59	58	imbi2d 330	. . . 4 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
60		breq2 4657	. . . . . . . . . 10
61		breq2 4657	. . . . . . . . . 10
62	60, 61	orbi12d 746	. . . . . . . . 9 $\/$ $\/$
63	62	3anbi1d 1403	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$
64	63	imbi1d 331	. . . . . . 7 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$
65	64	2ralbidv 2989	. . . . . 6 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
66	65	albidv 1849	. . . . 5 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
67	66	imbi2d 330	. . . 4 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
68	1	ad2antrr 762	. . . . . . . 8 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ Moore
69		mreexexlem2d.2	. . . . . . . 8 mrCls
70		mreexexlem2d.3	. . . . . . . 8 mrInd
71		mreexexlem2d.4	. . . . . . . . 9 $\$
72	71	ad2antrr 762	. . . . . . . 8 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
73		simplrl 800	. . . . . . . . 9 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
74	73	elpwid 4170	. . . . . . . 8 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
75		simplrr 801	. . . . . . . . 9 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
76	75	elpwid 4170	. . . . . . . 8 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
77		simpr2 1068	. . . . . . . 8 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$
78		simpr3 1069	. . . . . . . 8 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$
79		simpr1 1067	. . . . . . . . 9 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\/$
80		en0 8019	. . . . . . . . . 10
81		en0 8019	. . . . . . . . . 10
82	80, 81	orbi12i 543	. . . . . . . . 9 $\/$ $\/$
83	79, 82	sylib 208	. . . . . . . 8 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\/$
84	68, 69, 70, 72, 74, 76, 77, 78, 83	mreexexlem3d 16306	. . . . . . 7 $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $/\$
85	84	ex 450	. . . . . 6 $/\$ $\$ $/\$ $\$ $\/$ $/\$ $/\$ $/\$
86	85	ralrimivva 2971	. . . . 5 $\$ $\$ $\/$ $/\$ $/\$ $/\$
87	86	alrimiv 1855	. . . 4 $\$ $\$ $\/$ $/\$ $/\$ $/\$
88		nfv 1843	. . . . . . . . 9
89		nfv 1843	. . . . . . . . 9
90		nfa1 2028	. . . . . . . . 9 $\$ $\$ $\/$ $/\$ $/\$ $/\$
91	88, 89, 90	nf3an 1831	. . . . . . . 8 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
92		nfv 1843	. . . . . . . . . 10
93		nfv 1843	. . . . . . . . . 10
94		nfra1 2941	. . . . . . . . . . 11 $\$ $\$ $\/$ $/\$ $/\$ $/\$
95	94	nfal 2153	. . . . . . . . . 10 $\$ $\$ $\/$ $/\$ $/\$ $/\$
96	92, 93, 95	nf3an 1831	. . . . . . . . 9 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
97		nfv 1843	. . . . . . . . . . . . 13
98		nfv 1843	. . . . . . . . . . . . 13
99		nfra2 2946	. . . . . . . . . . . . . 14 $\$ $\$ $\/$ $/\$ $/\$ $/\$
100	99	nfal 2153	. . . . . . . . . . . . 13 $\$ $\$ $\/$ $/\$ $/\$ $/\$
101	97, 98, 100	nf3an 1831	. . . . . . . . . . . 12 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
102		nfv 1843	. . . . . . . . . . . 12 $\$
103	101, 102	nfan 1828	. . . . . . . . . . 11 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$
104	1	3ad2ant1 1082	. . . . . . . . . . . . . . 15 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ Moore
105	104	ad2antrr 762	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ Moore
106	71	3ad2ant1 1082	. . . . . . . . . . . . . . 15 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$
107	106	ad2antrr 762	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
108		simplrl 800	. . . . . . . . . . . . . . 15 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
109	108	elpwid 4170	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
110		simplrr 801	. . . . . . . . . . . . . . 15 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
111	110	elpwid 4170	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$
112		simpr2 1068	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$
113		simpr3 1069	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$
114		simpll2 1101	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$
115		simpll3 1102	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
116		simpr1 1067	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $\/$
117	105, 69, 70, 107, 109, 111, 112, 113, 114, 115, 116	mreexexlem4d 16307	. . . . . . . . . . . . 13 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $/\$ $\/$ $/\$ $/\$ $/\$
118	117	ex 450	. . . . . . . . . . . 12 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $\/$ $/\$ $/\$ $/\$
119	118	expr 643	. . . . . . . . . . 11 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
120	103, 119	ralrimi 2957	. . . . . . . . . 10 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
121	120	ex 450	. . . . . . . . 9 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
122	96, 121	ralrimi 2957	. . . . . . . 8 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
123	91, 122	alrimi 2082	. . . . . . 7 $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
124	123	3exp 1264	. . . . . 6 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
125	124	com12 32	. . . . 5 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
126	125	a2d 29	. . . 4 $\$ $\$ $\/$ $/\$ $/\$ $/\$ $\$ $\$ $\/$ $/\$ $/\$ $/\$
127	43, 51, 59, 67, 87, 126	finds 7092	. . 3 $\$ $\$ $\/$ $/\$ $/\$ $/\$
128	35, 127	mpcom 38	. 2 $\$ $\$ $\/$ $/\$ $/\$ $/\$
129	2, 3, 4, 5, 6, 26, 128	mreexexlemd 16304	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wal 1481

wceq 1483

wcel 1990

wral 2912

wrex 2913

cvv 3200 $\$ cdif 3571

cun 3572

cin 3573

wss 3574

c0 3915

cpw 4158

csn 4177 class class class wbr 4653

word 5722

csuc 5725

cfv 5888

com 7065

cen 7952

cfn 7955

ccrd 8761 Moorecmre 16242 mrClscmrc 16243 mrIndcmri 16244

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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-ac2 9285

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-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 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-int 4476 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-se 5074 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-isom 5897 df-riota 6611 df-om 7066 df-wrecs 7407 df-recs 7468 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-ac 8939 df-mre 16246 df-mrc 16247 df-mri 16248

This theorem is referenced by: (None)

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