Theorem elpr 3640

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 3639	. 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 2164  Vcvv 2760  {cpr 3620

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626

This theorem is referenced by:  prmg  3740  difprsnss  3757  preqr1  3795  preq12b  3797  prel12  3798  pwprss  3832  pwtpss  3833  unipr  3850  intpr  3903  zfpair2  4240  elop  4261  ordtri2or2exmidlem  4559  onsucelsucexmidlem  4562  en2lp  4587  reg3exmidlemwe  4612  xpsspw  4772  acexmidlem2  5916  2oconcl  6494  exmidpw  6966  exmidpweq  6967  renfdisj  8081  fzpr  10146  maxabslemval  11355  xrmaxiflemval  11396  isprm2  12258  2lgslem4  15260  bj-zfpair2  15472  ss1oel2o  15554