ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elpr2 GIF version

Theorem elpr2 3583
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, 14-Oct-2005.)
Hypotheses
Ref Expression
elpr2.1 𝐵 ∈ V
elpr2.2 𝐶 ∈ V
Assertion
Ref Expression
elpr2 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))

Proof of Theorem elpr2
StepHypRef Expression
1 elprg 3581 . . 3 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
21ibi 175 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
3 elpr2.1 . . . . . 6 𝐵 ∈ V
4 eleq1 2220 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
53, 4mpbiri 167 . . . . 5 (𝐴 = 𝐵𝐴 ∈ V)
6 elpr2.2 . . . . . 6 𝐶 ∈ V
7 eleq1 2220 . . . . . 6 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
86, 7mpbiri 167 . . . . 5 (𝐴 = 𝐶𝐴 ∈ V)
95, 8jaoi 706 . . . 4 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ V)
10 elprg 3581 . . . 4 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
119, 10syl 14 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
1211ibir 176 . 2 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
132, 12impbii 125 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wb 104  wo 698   = wceq 1335  wcel 2128  Vcvv 2712  {cpr 3562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3567  df-pr 3568
This theorem is referenced by:  elxr  9690
  Copyright terms: Public domain W3C validator