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Theorem elprg 3363
 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 2024 . . 3 (x = A → (x = BA = B))
2 eqeq1 2024 . . 3 (x = A → (x = 𝐶A = 𝐶))
31, 2orbi12d 694 . 2 (x = A → ((x = B x = 𝐶) ↔ (A = B A = 𝐶)))
4 dfpr2 3362 . 2 {B, 𝐶} = {x ∣ (x = B x = 𝐶)}
53, 4elab2g 2662 1 (A 𝑉 → (A {B, 𝐶} ↔ (A = B A = 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 616   = wceq 1226   ∈ wcel 1370  {cpr 3347 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-sn 3352  df-pr 3353 This theorem is referenced by:  elpr  3364  elpr2  3365  elpri  3366  eltpg  3386  prid1g  3444  preqr1g  3507
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