Theorem elpr 3729

Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Hypothesis

Ref	Expression
elpr.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elpr	⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))

Proof of Theorem elpr

Step	Hyp	Ref	Expression
1		elpr.1	. 2 ⊢ 𝐴 ∈ V
2		elprg 3728	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))

Syntax hints:   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {cpr 3709

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715

This theorem is referenced by:  prmg  3833  difprsnss  3851  preqr1  3891  preq12b  3893  prel12  3894  pwprss  3929  pwtpss  3930  unipr  3947  intpr  4000  zfpair2  4345  elop  4369  ordtri2or2exmidlem  4671  onsucelsucexmidlem  4674  en2lp  4699  reg3exmidlemwe  4724  xpsspw  4885  acexmidlem2  6075  2oconcl  6705  exmidpw  7208  exmidpweq  7209  renfdisj  8378  fzpr  10465  maxabslemval  11955  xrmaxiflemval  11997  isprm2  12876  2lgslem4  16139  structiedg0val  16198  bj-zfpair2  16853  ss1oel2o  16934