Theorem elpr 3644

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 3643	. 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 709   = wceq 1364   ∈ wcel 2167  Vcvv 2763  {cpr 3624

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630

This theorem is referenced by:  prmg  3744  difprsnss  3761  preqr1  3799  preq12b  3801  prel12  3802  pwprss  3836  pwtpss  3837  unipr  3854  intpr  3907  zfpair2  4244  elop  4265  ordtri2or2exmidlem  4563  onsucelsucexmidlem  4566  en2lp  4591  reg3exmidlemwe  4616  xpsspw  4776  acexmidlem2  5922  2oconcl  6506  exmidpw  6978  exmidpweq  6979  renfdisj  8103  fzpr  10169  maxabslemval  11390  xrmaxiflemval  11432  isprm2  12310  2lgslem4  15428  bj-zfpair2  15640  ss1oel2o  15722