Theorem elpr 3712

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 3711	. 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 716   = wceq 1398   ∈ wcel 2205  Vcvv 2815  {cpr 3692

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698

This theorem is referenced by:  prmg  3816  difprsnss  3834  preqr1  3874  preq12b  3876  prel12  3877  pwprss  3912  pwtpss  3913  unipr  3930  intpr  3983  zfpair2  4325  elop  4349  ordtri2or2exmidlem  4650  onsucelsucexmidlem  4653  en2lp  4678  reg3exmidlemwe  4703  xpsspw  4864  acexmidlem2  6049  2oconcl  6674  exmidpw  7170  exmidpweq  7171  renfdisj  8335  fzpr  10415  maxabslemval  11897  xrmaxiflemval  11939  isprm2  12818  2lgslem4  15993  structiedg0val  16052  bj-zfpair2  16697  ss1oel2o  16778