Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pr1nebg Structured version   Visualization version   GIF version

Theorem pr1nebg 4751
 Description: A (proper) pair is not equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
Assertion
Ref Expression
pr1nebg (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴𝐶 ↔ {𝐴, 𝐵} ≠ {𝐵, 𝐶}))

Proof of Theorem pr1nebg
StepHypRef Expression
1 pr1eqbg 4750 . 2 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))
21necon3bid 3031 1 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴𝐶 ↔ {𝐴, 𝐵} ≠ {𝐵, 𝐶}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   ≠ wne 2987  {cpr 4530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3444  df-un 3888  df-sn 4529  df-pr 4531 This theorem is referenced by:  usgr2pthlem  27596
 Copyright terms: Public domain W3C validator