rtrclex - Mathbox for Richard Penner

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Theorem rtrclex 37924

Description: The reflexive-transitive closure of a set exists. (Contributed by RP, 1-Nov-2020.)

**Assertion**
Ref	Expression
rtrclex	$/\$ $/\$

Distinct variable group:

**Proof of Theorem rtrclex**
Step	Hyp	Ref	Expression
1		ssun1 3776	. . . 4
2		coundir 5637	. . . . . . 7
3		coundi 5636	. . . . . . . . 9
4		cossxp 5658	. . . . . . . . . . 11
5		ssun1 3776	. . . . . . . . . . . 12
6		ssun2 3777	. . . . . . . . . . . 12
7		xpss12 5225	. . . . . . . . . . . 12 $/\$
8	5, 6, 7	mp2an 708	. . . . . . . . . . 11
9	4, 8	sstri 3612	. . . . . . . . . 10
10		cossxp 5658	. . . . . . . . . . 11
11		dmxpss 5565	. . . . . . . . . . . 12
12		xpss12 5225	. . . . . . . . . . . 12 $/\$
13	11, 6, 12	mp2an 708	. . . . . . . . . . 11
14	10, 13	sstri 3612	. . . . . . . . . 10
15	9, 14	unssi 3788	. . . . . . . . 9
16	3, 15	eqsstri 3635	. . . . . . . 8
17		coundi 5636	. . . . . . . . 9
18		cossxp 5658	. . . . . . . . . . 11
19		rnxpss 5566	. . . . . . . . . . . 12
20		xpss12 5225	. . . . . . . . . . . 12 $/\$
21	5, 19, 20	mp2an 708	. . . . . . . . . . 11
22	18, 21	sstri 3612	. . . . . . . . . 10
23		xpidtr 5518	. . . . . . . . . 10
24	22, 23	unssi 3788	. . . . . . . . 9
25	17, 24	eqsstri 3635	. . . . . . . 8
26	16, 25	unssi 3788	. . . . . . 7
27	2, 26	eqsstri 3635	. . . . . 6
28		ssun2 3777	. . . . . 6
29	27, 28	sstri 3612	. . . . 5
30		dmun 5331	. . . . . . . . . . 11
31		dmxpid 5345	. . . . . . . . . . . . 13
32	31	uneq2i 3764	. . . . . . . . . . . 12
33		ssequn1 3783	. . . . . . . . . . . . 13
34	5, 33	mpbi 220	. . . . . . . . . . . 12
35	32, 34	eqtri 2644	. . . . . . . . . . 11
36	30, 35	eqtri 2644	. . . . . . . . . 10
37		rnun 5541	. . . . . . . . . . 11
38		rnxpid 5567	. . . . . . . . . . . . 13
39	38	uneq2i 3764	. . . . . . . . . . . 12
40		ssequn1 3783	. . . . . . . . . . . . 13
41	6, 40	mpbi 220	. . . . . . . . . . . 12
42	39, 41	eqtri 2644	. . . . . . . . . . 11
43	37, 42	eqtri 2644	. . . . . . . . . 10
44	36, 43	uneq12i 3765	. . . . . . . . 9
45		unidm 3756	. . . . . . . . 9
46	44, 45	eqtri 2644	. . . . . . . 8
47	46	reseq2i 5393	. . . . . . 7
48		fnresi 6008	. . . . . . . . 9
49		fnrel 5989	. . . . . . . . 9
50		relssdmrn 5656	. . . . . . . . 9
51	48, 49, 50	mp2b 10	. . . . . . . 8
52		dmresi 5457	. . . . . . . . 9
53		rnresi 5479	. . . . . . . . 9
54	52, 53	xpeq12i 5137	. . . . . . . 8
55	51, 54	sseqtri 3637	. . . . . . 7
56	47, 55	eqsstri 3635	. . . . . 6
57	56, 28	sstri 3612	. . . . 5
58	29, 57	pm3.2i 471	. . . 4 $/\$
59		rtrclexlem 37923	. . . . 5
60		id 22	. . . . . . . . . . 11
61	60, 60	coeq12d 5286	. . . . . . . . . 10
62	61, 60	sseq12d 3634	. . . . . . . . 9
63		dmeq 5324	. . . . . . . . . . . 12
64		rneq 5351	. . . . . . . . . . . 12
65	63, 64	uneq12d 3768	. . . . . . . . . . 11
66	65	reseq2d 5396	. . . . . . . . . 10
67	66, 60	sseq12d 3634	. . . . . . . . 9
68	62, 67	anbi12d 747	. . . . . . . 8 $/\$ $/\$
69	68	cleq2lem 37914	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
70	69	biimprd 238	. . . . . 6 $/\$ $/\$ $/\$ $/\$
71	70	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
72	59, 71	spcimedv 3292	. . . 4 $/\$ $/\$ $/\$ $/\$
73	1, 58, 72	mp2ani 714	. . 3 $/\$ $/\$
74		exsimpl 1795	. . . 4 $/\$ $/\$
75		vex 3203	. . . . . 6
76	75	ssex 4802	. . . . 5
77	76	exlimiv 1858	. . . 4
78	74, 77	syl 17	. . 3 $/\$ $/\$
79	73, 78	impbii 199	. 2 $/\$ $/\$
80		intexab 4822	. 2 $/\$ $/\$ $/\$ $/\$
81	79, 80	bitri 264	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wex 1704

wcel 1990

cab 2608

cvv 3200

cun 3572

wss 3574

cint 4475

cid 5023

cxp 5112

cdm 5114

crn 5115

cres 5116

ccom 5118

wrel 5119

wfn 5883

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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-id 5024 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-fun 5890 df-fn 5891

This theorem is referenced by: (None)

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