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Theorem elsuci 5829
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 5767 . . . 4 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2722 . . 3 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 3786 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
42, 3bitri 264 . 2 (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
5 elsni 4227 . . 3 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
65orim2i 539 . 2 ((𝐴𝐵𝐴 ∈ {𝐵}) → (𝐴𝐵𝐴 = 𝐵))
74, 6sylbi 207 1 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382   = wceq 1523  wcel 2030  cun 3605  {csn 4210  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-sn 4211  df-suc 5767
This theorem is referenced by:  suctr  5846  trsucss  5849  ordnbtwn  5854  ordnbtwnOLD  5855  suc11  5869  tfrlem11  7529  omordi  7691  nnmordi  7756  phplem3  8182  pssnn  8219  r1sdom  8675  cfsuc  9117  axdc3lem2  9311  axdc3lem4  9313  indpi  9767  bnj563  30939  bnj964  31139  ontgval  32555  onsucconni  32561  suctrALT  39375  suctrALT2VD  39385  suctrALT2  39386  suctrALTcf  39472  suctrALTcfVD  39473  suctrALT3  39474
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