Theorem elpr 3730

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 3729	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))

This proof depends on syntax axioms:   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {cpr 3710

This proof depends on 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 proof 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 3715  df-pr 3716

This theorem is used by:  prmg  3835  difprsnss  3853  preqr1  3893  preq12b  3895  prel12  3896  pwprss  3931  pwtpss  3932  unipr  3949  intpr  4002  zfpair2  4347  elop  4371  ordtri2or2exmidlem  4673  onsucelsucexmidlem  4676  en2lp  4701  reg3exmidlemwe  4726  xpsspw  4887  acexmidlem2  6082  2oconcl  6712  exmidpw  7215  exmidpweq  7216  renfdisj  8385  fzpr  10484  maxabslemval  11974  xrmaxiflemval  12016  isprm2  12895  2lgslem4  16222  structiedg0val  16281  bj-zfpair2  16936  ss1oel2o  17017