Theorem 0rngo 38565

Description: Obsolete theorem, use 0ring01eqbi2 20615 instead. In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
0ring.1	⊢ 𝐺 = (1st ‘𝑅)
0ring.2	⊢ 𝐻 = (2nd ‘𝑅)
0ring.3	⊢ 𝑋 = ran 𝐺
0ring.4	⊢ 𝑍 = (GId‘𝐺)
0ring.5	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
0rngo	⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))

Proof of Theorem 0rngo

Step	Hyp	Ref	Expression
1		0ring.4	. . . . . . 7 ⊢ 𝑍 = (GId‘𝐺)
2	1	fvexi 6896	. . . . . 6 ⊢ 𝑍 ∈ V
3	2	snid 4633	. . . . 5 ⊢ 𝑍 ∈ {𝑍}
4		eleq1 2857	. . . . 5 ⊢ (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
5	3, 4	mpbii 236	. . . 4 ⊢ (𝑍 = 𝑈 → 𝑈 ∈ {𝑍})
6		0ring.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
7	6, 1	0idl 38563	. . . . 5 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
8		0ring.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
9		0ring.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
10		0ring.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
11	6, 8, 9, 10	1idl 38564	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
12	7, 11	mpdan 699	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
13	5, 12	imbitrid 247	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
14		eqcom 2776	. . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
15	13, 14	imbitrdi 254	. 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍}))
16	6	rneqi 5928	. . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅)
17	9, 16	eqtri 2792	. . . 4 ⊢ 𝑋 = ran (1st ‘𝑅)
18	17, 8, 10	rngo1cl 38477	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
19		eleq2 2858	. . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
20		elsni 4611	. . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
21	20	eqcomd 2775	. . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
22	19, 21	biimtrdi 256	. . 3 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑍 = 𝑈))
23	18, 22	syl5com 32	. 2 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
24	15, 23	impbid 215	1 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  {csn 4594  ran crn 5663  ‘cfv 6537  1^st c1st 7983  2^nd c2nd 7984  GIdcgi 30782  RingOpscrngo 38432  Idlcidl 38545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-1st 7985  df-2nd 7986  df-grpo 30785  df-gid 30786  df-ginv 30787  df-ablo 30837  df-ass 38381  df-exid 38383  df-mgmOLD 38387  df-sgrOLD 38399  df-mndo 38405  df-rngo 38433  df-idl 38548

This theorem is referenced by:  smprngopr  38590  isfldidl2  38607