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Theorem elsuci 4362
 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 4330 . . . 4 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2224 . . 3 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 3248 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
42, 3bitri 183 . 2 (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
5 elsni 3578 . . 3 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
65orim2i 751 . 2 ((𝐴𝐵𝐴 ∈ {𝐵}) → (𝐴𝐵𝐴 = 𝐵))
74, 6sylbi 120 1 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 698   = wceq 1335   ∈ wcel 2128   ∪ cun 3100  {csn 3560  suc csuc 4324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-suc 4330 This theorem is referenced by:  trsucss  4382  onsucelsucexmid  4487  ordsoexmid  4519  ordsuc  4520  ordpwsucexmid  4527  nnsucelsuc  6431  nntri3or  6433  nnmordi  6456  nnaordex  6467  phplem3  6792  nnnninf2  7059  3nelsucpw1  7152  3nsssucpw1  7154
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