Theorem elop 4153

Description: An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
elop.1	⊢ 𝐴 ∈ V
elop.2	⊢ 𝐵 ∈ V
elop.3	⊢ 𝐶 ∈ V

Assertion

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

Proof of Theorem elop

Step	Hyp	Ref	Expression
1		elop.2	. . . 4 ⊢ 𝐵 ∈ V
2		elop.3	. . . 4 ⊢ 𝐶 ∈ V
3	1, 2	dfop 3704	. . 3 ⊢ ⟨𝐵, 𝐶⟩ = {{𝐵}, {𝐵, 𝐶}}
4	3	eleq2i 2206	. 2 ⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}})
5		elop.1	. . 3 ⊢ 𝐴 ∈ V
6	5	elpr 3548	. 2 ⊢ (𝐴 ∈ {{𝐵}, {𝐵, 𝐶}} ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
7	4, 6	bitri 183	1 ⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))

Syntax hints:   ↔ wb 104   ∨ wo 697   = wceq 1331   ∈ wcel 1480  Vcvv 2686  {csn 3527  {cpr 3528  ⟨cop 3530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536

This theorem is referenced by:  relop  4689  bdop  13073