Theorem elsuci 4506

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 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))

Proof of Theorem elsuci

Step	Hyp	Ref	Expression
1		df-suc 4474	. . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2	1	eleq2i 2298	. . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵}))
3		elun 3350	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
4	2, 3	bitri 184	. 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
5		elsni 3691	. . 3 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
6	5	orim2i 769	. 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
7	4, 6	sylbi 121	1 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∨ wo 716   = wceq 1398   ∈ wcel 2202   ∪ cun 3199  {csn 3673  suc csuc 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-suc 4474

This theorem is referenced by:  trsucss  4526  onsucelsucexmid  4634  ordsoexmid  4666  ordsuc  4667  ordpwsucexmid  4674  nnsucelsuc  6702  nntri3or  6704  nnmordi  6727  nnaordex  6739  phplem3  7083  nninfninc  7365  nnnninf2  7369  3nelsucpw1  7495  3nsssucpw1  7497