Theorem elsuci 6429

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 6368	. . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2	1	eleq2i 2826	. . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵}))
3		elun 4148	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
4	2, 3	bitri 275	. 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
5		elsni 4645	. . 3 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
6	5	orim2i 910	. 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
7	4, 6	sylbi 216	1 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1542   ∈ wcel 2107   ∪ cun 3946  {csn 4628  suc csuc 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3953  df-sn 4629  df-suc 6368

This theorem is referenced by:  suctr  6448  trsucss  6450  ordnbtwn  6455  suc11  6469  tfrlem11  8385  omordi  8563  nnmordi  8628  pssnn  9165  phplem3OLD  9216  pssnnOLD  9262  r1sdom  9766  cfsuc  10249  axdc3lem2  10443  axdc3lem4  10445  indpi  10899  bnj563  33743  bnj964  33943  ontgval  35305  onsucconni  35311  suctrALT  43573  suctrALT2VD  43583  suctrALT2  43584  suctrALTcf  43669  suctrALTcfVD  43670  suctrALT3  43671