Theorem elpr 3716

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 3715	. 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 3696

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 3218  df-sn 3701  df-pr 3702

This theorem is referenced by:  prmg  3820  difprsnss  3838  preqr1  3878  preq12b  3880  prel12  3881  pwprss  3916  pwtpss  3917  unipr  3934  intpr  3987  zfpair2  4329  elop  4353  ordtri2or2exmidlem  4655  onsucelsucexmidlem  4658  en2lp  4683  reg3exmidlemwe  4708  xpsspw  4869  acexmidlem2  6057  2oconcl  6687  exmidpw  7183  exmidpweq  7184  renfdisj  8351  fzpr  10438  maxabslemval  11924  xrmaxiflemval  11966  isprm2  12845  2lgslem4  16108  structiedg0val  16167  bj-zfpair2  16822  ss1oel2o  16903