Theorem elsuci 6375

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 6312	. . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2	1	eleq2i 2823	. . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵}))
3		elun 4100	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
4	2, 3	bitri 275	. 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
5		elsni 4590	. . 3 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
6	5	orim2i 910	. 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 847   = wceq 1541   ∈ wcel 2111   ∪ cun 3895  {csn 4573  suc csuc 6308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-sn 4574  df-suc 6312

This theorem is referenced by:  suctr  6394  trsucss  6396  ordnbtwn  6401  suc11  6415  tfrlem11  8307  omordi  8481  nnmordi  8546  pssnn  9078  r1sdom  9667  cfsuc  10148  axdc3lem2  10342  axdc3lem4  10344  indpi  10798  constrmon  33757  bnj563  34755  bnj964  34955  ontgval  36475  onsucconni  36481  suctrALT  44917  suctrALT2VD  44927  suctrALT2  44928  suctrALTcf  45013  suctrALTcfVD  45014  suctrALT3  45015