Theorem 0ring 16175

Description: In a ring, 0 = 1 iff the ring contains only 0.

Hypotheses

Ref	Expression
0ring.1	|- G = (1st` R)
0ring.2	|- H = (2nd` R)
0ring.3	|- X = ran G
0ring.4	|- Z = (Id` G)
0ring.5	|- U = (Id` H)

Assertion

Ref	Expression
0ring	|- (R e. Ring -> (Z = U <-> X = {Z}))

Proof of Theorem 0ring

Step	Hyp	Ref	Expression
1		0ring.1	. . . . . 6 |- G = (1st` R)
2		0ring.4	. . . . . 6 |- Z = (Id` G)
3	1, 2	0idl 16173	. . . . 5 |- (R e. Ring -> {Z} e. (Idl` R))
4		0ring.2	. . . . . 6 |- H = (2nd` R)
5		0ring.3	. . . . . 6 |- X = ran G
6		0ring.5	. . . . . 6 |- U = (Id` H)
7	1, 4, 5, 6	1idl 16174	. . . . 5 |- ((R e. Ring /\ {Z} e. (Idl` R)) -> (U e. {Z} <-> {Z} = X))
8	3, 7	mpdan 768	. . . 4 |- (R e. Ring -> (U e. {Z} <-> {Z} = X))
9		fvex 4689	. . . . . . 7 |- (Id` G) e. _V
10	2, 9	eqeltri 1967	. . . . . 6 |- Z e. _V
11	10	snid 3069	. . . . 5 |- Z e. {Z}
12		eleq1 1957	. . . . 5 |- (Z = U -> (Z e. {Z} <-> U e. {Z}))
13	11, 12	mpbii 210	. . . 4 |- (Z = U -> U e. {Z})
14	8, 13	syl5bi 225	. . 3 |- (R e. Ring -> (Z = U -> {Z} = X))
15		eqcom 1886	. . 3 |- ({Z} = X <-> X = {Z})
16	14, 15	syl6ib 229	. 2 |- (R e. Ring -> (Z = U -> X = {Z}))
17		eleq2 1958	. . . 4 |- (X = {Z} -> (U e. X <-> U e. {Z}))
18		elsni 3066	. . . . 5 |- (U e. {Z} -> U = Z)
19	18	eqcomd 1889	. . . 4 |- (U e. {Z} -> Z = U)
20	17, 19	syl6bi 231	. . 3 |- (X = {Z} -> (U e. X -> Z = U))
21	1	rneqi 4187	. . . . 5 |- ran G = ran (1st` R)
22	5, 21	eqtri 1908	. . . 4 |- X = ran (1st` R)
23	22, 4, 6	ring1cl 10415	. . 3 |- (R e. Ring -> U e. X)
24	20, 23	syl5com 63	. 2 |- (R e. Ring -> (X = {Z} -> Z = U))
25	16, 24	impbid 574	1 |- (R e. Ring -> (Z = U <-> X = {Z}))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044  ran crn 3987  ` cfv 3998  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463  Idlcidl 16155

This theorem is referenced by:  smprngpr 16200  isfldidl2 16217

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-ring 9464  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385  df-idl 16158

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