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Theorem 0rngo 38035
Description: In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.)
Hypotheses
Ref Expression
0ring.1 𝐺 = (1st𝑅)
0ring.2 𝐻 = (2nd𝑅)
0ring.3 𝑋 = ran 𝐺
0ring.4 𝑍 = (GId‘𝐺)
0ring.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
0rngo (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))

Proof of Theorem 0rngo
StepHypRef Expression
1 0ring.4 . . . . . . 7 𝑍 = (GId‘𝐺)
21fvexi 6919 . . . . . 6 𝑍 ∈ V
32snid 4661 . . . . 5 𝑍 ∈ {𝑍}
4 eleq1 2828 . . . . 5 (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
53, 4mpbii 233 . . . 4 (𝑍 = 𝑈𝑈 ∈ {𝑍})
6 0ring.1 . . . . . 6 𝐺 = (1st𝑅)
76, 10idl 38033 . . . . 5 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
8 0ring.2 . . . . . 6 𝐻 = (2nd𝑅)
9 0ring.3 . . . . . 6 𝑋 = ran 𝐺
10 0ring.5 . . . . . 6 𝑈 = (GId‘𝐻)
116, 8, 9, 101idl 38034 . . . . 5 ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
127, 11mpdan 687 . . . 4 (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
135, 12imbitrid 244 . . 3 (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
14 eqcom 2743 . . 3 ({𝑍} = 𝑋𝑋 = {𝑍})
1513, 14imbitrdi 251 . 2 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
166rneqi 5947 . . . . 5 ran 𝐺 = ran (1st𝑅)
179, 16eqtri 2764 . . . 4 𝑋 = ran (1st𝑅)
1817, 8, 10rngo1cl 37947 . . 3 (𝑅 ∈ RingOps → 𝑈𝑋)
19 eleq2 2829 . . . 4 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
20 elsni 4642 . . . . 5 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2120eqcomd 2742 . . . 4 (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
2219, 21biimtrdi 253 . . 3 (𝑋 = {𝑍} → (𝑈𝑋𝑍 = 𝑈))
2318, 22syl5com 31 . 2 (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
2415, 23impbid 212 1 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  {csn 4625  ran crn 5685  cfv 6560  1st c1st 8013  2nd c2nd 8014  GIdcgi 30510  RingOpscrngo 37902  Idlcidl 38015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-1st 8015  df-2nd 8016  df-grpo 30513  df-gid 30514  df-ginv 30515  df-ablo 30565  df-ass 37851  df-exid 37853  df-mgmOLD 37857  df-sgrOLD 37869  df-mndo 37875  df-rngo 37903  df-idl 38018
This theorem is referenced by:  smprngopr  38060  isfldidl2  38077
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