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Theorem elop 3959
 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
StepHypRef Expression
1 elop.2 . . . 4 B V
2 elop.3 . . . 4 𝐶 V
31, 2dfop 3539 . . 3 B, 𝐶⟩ = {{B}, {B, 𝐶}}
43eleq2i 2101 . 2 (A B, 𝐶⟩ ↔ A {{B}, {B, 𝐶}})
5 elop.1 . . 3 A V
65elpr 3385 . 2 (A {{B}, {B, 𝐶}} ↔ (A = {B} A = {B, 𝐶}))
74, 6bitri 173 1 (A B, 𝐶⟩ ↔ (A = {B} A = {B, 𝐶}))
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367  {cpr 3368  ⟨cop 3370 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376 This theorem is referenced by:  relop  4429  bdop  9310
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