Theorem 0rngo 38961

Description: Obsolete theorem, use 0ring01eqbi2 20783 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 6899	. . . . . 6 ⊢ 𝑍 ∈ V
3	2	snid 4623	. . . . 5 ⊢ 𝑍 ∈ {𝑍}
4		eleq1 2849	. . . . 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 38959	. . . . 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 38960	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
12	7, 11	mpdan 700	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
13	5, 12	imbitrid 247	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
14		eqcom 2768	. . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
15	13, 14	imbitrdi 254	. 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍}))
16	6	rneqi 5919	. . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅)
17	9, 16	eqtri 2784	. . . 4 ⊢ 𝑋 = ran (1st ‘𝑅)
18	17, 8, 10	rngo1cl 38873	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
19		eleq2 2850	. . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
20		elsni 4601	. . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
21	20	eqcomd 2767	. . . 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}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {csn 4584  ran crn 5652  ‘cfv 6538  1^st c1st 7999  2^nd c2nd 8000  GIdcgi 31092  RingOpscrngo 38828  Idlcidl 38941

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-1st 8001  df-2nd 8002  df-grpo 31095  df-gid 31096  df-ginv 31097  df-ablo 31147  df-ass 38777  df-exid 38779  df-mgmOLD 38783  df-sgrOLD 38795  df-mndo 38801  df-rngo 38829  df-idl 38944

This theorem is used by:  smprngopr  38986  isfldidl2  39003