Theorem elsuci 6385

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 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))

Proof of Theorem elsuci

Step	Hyp	Ref	Expression
1		df-suc 6322	. . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2	1	eleq2i 2827	. . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵}))
3		elun 4104	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
4	2, 3	bitri 275	. 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
5		elsni 4596	. . 3 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
6	5	orim2i 911	. 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
7	4, 6	sylbi 217	1 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∪ cun 3898  {csn 4579  suc csuc 6318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3441  df-un 3905  df-sn 4580  df-suc 6322

This theorem is referenced by:  suctr  6404  trsucss  6406  ordnbtwn  6411  suc11  6425  tfrlem11  8319  omordi  8493  nnmordi  8559  pssnn  9095  r1sdom  9688  cfsuc  10169  axdc3lem2  10363  axdc3lem4  10365  indpi  10820  constrmon  33880  bnj563  34878  bnj964  35078  ontgval  36604  onsucconni  36610  suctrALT  45103  suctrALT2VD  45113  suctrALT2  45114  suctrALTcf  45199  suctrALTcfVD  45200  suctrALT3  45201