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Theorem elprg 3384
 Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
Assertion
Ref Expression
elprg (A 𝑉 → (A {B, 𝐶} ↔ (A = B A = 𝐶)))

Proof of Theorem elprg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . 3 (x = A → (x = BA = B))
2 eqeq1 2043 . . 3 (x = A → (x = 𝐶A = 𝐶))
31, 2orbi12d 706 . 2 (x = A → ((x = B x = 𝐶) ↔ (A = B A = 𝐶)))
4 dfpr2 3383 . 2 {B, 𝐶} = {x ∣ (x = B x = 𝐶)}
53, 4elab2g 2683 1 (A 𝑉 → (A {B, 𝐶} ↔ (A = B A = 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390  {cpr 3368 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-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 This theorem is referenced by:  elpr  3385  elpr2  3386  elpri  3387  eltpg  3407  prid1g  3465  preqr1g  3528
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