Theorem elop 5437

Description: Characterization of the elements of an ordered pair. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) Remove an extraneous hypothesis. (Revised by BJ, 25-Dec-2020.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
elop.1	⊢ 𝐵 ∈ V
elop.2	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
elop	⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))

Proof of Theorem elop

Step	Hyp	Ref	Expression
1		elop.1	. 2 ⊢ 𝐵 ∈ V
2		elop.2	. 2 ⊢ 𝐶 ∈ V
3		elopg 5436	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})))
4	1, 2, 3	mp2an 702	1 ⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))

Syntax hints:   ↔ wb 208   ∨ wo 858   = wceq 1562   ∈ wcel 2144  Vcvv 3456  {csn 4584  {cpr 4586  ⟨cop 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591

This theorem is referenced by:  relop  5824