Theorem elsuci 2289

Description: Membership in a successor. This one-way implication does not require that either A or B be sets.

Assertion

Ref	Expression
elsuci	⊢ (A ∈ suc B → (A ∈ B ∨ A = B))

Proof of Theorem elsuci

Step	Hyp	Ref	Expression
1		df-suc 2205	. . . 4 ⊢ suc B = (B ∪ {B})
2	1	eleq2i 1153	. . 3 ⊢ (A ∈ suc B ↔ A ∈ (B ∪ {B}))
3		elun 1601	. . 3 ⊢ (A ∈ (B ∪ {B}) ↔ (A ∈ B ∨ A ∈ {B}))
4	2, 3	bitr 151	. 2 ⊢ (A ∈ suc B ↔ (A ∈ B ∨ A ∈ {B}))
5		elsni 1827	. . 3 ⊢ (A ∈ {B} → A = B)
6	5	orim2i 273	. 2 ⊢ ((A ∈ B ∨ A ∈ {B}) → (A ∈ B ∨ A = B))
7	4, 6	sylbi 174	1 ⊢ (A ∈ suc B → (A ∈ B ∨ A = B))

Syntax hints:   → wi 2   ∨ wo 195   = wceq 1091   ∈ wcel 1092   ∪ cun 1485  {csn 1808  suc csuc 2201

This theorem is referenced by:  trsucss 2309  ordnbtwn 2316  suc11 2341  tfrlem11 2959  omordi 3164  phplem4 3406  pssnn 3428  cfsuc 3709  indpi 3828

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-suc 2205

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