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Theorem elsuci 6453
Description: Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. Lemma 1.13 of [Schloeder] p. 2. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
elsuci (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem elsuci
StepHypRef Expression
1 df-suc 6392 . . . 4 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2831 . . 3 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 4163 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
42, 3bitri 275 . 2 (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
5 elsni 4648 . . 3 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
65orim2i 910 . 2 ((𝐴𝐵𝐴 ∈ {𝐵}) → (𝐴𝐵𝐴 = 𝐵))
74, 6sylbi 217 1 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 847   = wceq 1537  wcel 2106  cun 3961  {csn 4631  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-suc 6392
This theorem is referenced by:  suctr  6472  trsucss  6474  ordnbtwn  6479  suc11  6493  tfrlem11  8427  omordi  8603  nnmordi  8668  pssnn  9207  phplem3OLD  9254  r1sdom  9812  cfsuc  10295  axdc3lem2  10489  axdc3lem4  10491  indpi  10945  constrmon  33749  bnj563  34736  bnj964  34936  ontgval  36414  onsucconni  36420  suctrALT  44824  suctrALT2VD  44834  suctrALT2  44835  suctrALTcf  44920  suctrALTcfVD  44921  suctrALT3  44922
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