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Theorem elsuci 5708
 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 5646 . . . 4 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2680 . . 3 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 3715 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
42, 3bitri 263 . 2 (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
5 elsni 4142 . . 3 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
65orim2i 539 . 2 ((𝐴𝐵𝐴 ∈ {𝐵}) → (𝐴𝐵𝐴 = 𝐵))
74, 6sylbi 206 1 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  {csn 4125  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-suc 5646 This theorem is referenced by:  suctr  5725  trsucss  5728  ordnbtwn  5733  ordnbtwnOLD  5734  suc11  5748  tfrlem11  7371  omordi  7533  nnmordi  7598  phplem3  8026  pssnn  8063  r1sdom  8520  cfsuc  8962  axdc3lem2  9156  axdc3lem4  9158  indpi  9608  bnj563  30067  bnj964  30267  ontgval  31600  onsucconi  31606  suctrALT  38083  suctrALT2VD  38093  suctrALT2  38094  suctrALTcf  38180  suctrALTcfVD  38181  suctrALT3  38182
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