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Theorem elsuci 4187
 Description: Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
elsuci (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem elsuci
StepHypRef Expression
1 df-suc 4155 . . . 4 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2149 . . 3 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 3124 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
42, 3bitri 182 . 2 (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
5 elsni 3435 . . 3 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
65orim2i 711 . 2 ((𝐴𝐵𝐴 ∈ {𝐵}) → (𝐴𝐵𝐴 = 𝐵))
74, 6sylbi 119 1 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 662   = wceq 1285   ∈ wcel 1434   ∪ cun 2981  {csn 3417  suc csuc 4149 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-un 2987  df-sn 3423  df-suc 4155 This theorem is referenced by:  trsucss  4207  onsucelsucexmid  4302  ordsoexmid  4334  ordsuc  4335  ordpwsucexmid  4342  nnsucelsuc  6157  nntri3or  6159  nnmordi  6178  nnaordex  6189  phplem3  6412
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