Theorem elsuci 4106

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 4074	. . . 4 ⊢ suc B = (B ∪ {B})
2	1	eleq2i 2101	. . 3 ⊢ (A ∈ suc B ↔ A ∈ (B ∪ {B}))
3		elun 3078	. . 3 ⊢ (A ∈ (B ∪ {B}) ↔ (A ∈ B ∨ A ∈ {B}))
4	2, 3	bitri 173	. 2 ⊢ (A ∈ suc B ↔ (A ∈ B ∨ A ∈ {B}))
5		elsni 3391	. . 3 ⊢ (A ∈ {B} → A = B)
6	5	orim2i 677	. 2 ⊢ ((A ∈ B ∨ A ∈ {B}) → (A ∈ B ∨ A = B))
7	4, 6	sylbi 114	1 ⊢ (A ∈ suc B → (A ∈ B ∨ A = B))

Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ∪ cun 2909  {csn 3367  suc csuc 4068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-suc 4074

This theorem is referenced by:  trsucss  4126  onsucelsucexmid  4215  ordsoexmid  4240  ordsuc  4241  ordpwsucexmid  4246  nnsucelsuc  6009  nntri3or  6011  nnmordi  6025  nnaordex  6036