Theorem elop 4275

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 3818	. . 3 ⊢ ⟨𝐵, 𝐶⟩ = {{𝐵}, {𝐵, 𝐶}}
4	3	eleq2i 2272	. 2 ⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}})
5		elop.1	. . 3 ⊢ 𝐴 ∈ V
6	5	elpr 3654	. 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 710   = wceq 1373   ∈ wcel 2176  Vcvv 2772  {csn 3633  {cpr 3634  ⟨cop 3636

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642

This theorem is referenced by:  relop  4828  funopsn  5762  bdop  15811