Theorem elop 4260

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 3803	. . 3 ⊢ ⟨𝐵, 𝐶⟩ = {{𝐵}, {𝐵, 𝐶}}
4	3	eleq2i 2260	. 2 ⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}})
5		elop.1	. . 3 ⊢ 𝐴 ∈ V
6	5	elpr 3639	. 2 ⊢ (𝐴 ∈ {{𝐵}, {𝐵, 𝐶}} ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
7	4, 6	bitri 184	1 ⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))

Syntax hints:   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2164  Vcvv 2760  {csn 3618  {cpr 3619  ⟨cop 3621

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627

This theorem is referenced by:  relop  4812  bdop  15367