MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prprc1 Structured version   Visualization version   GIF version

Theorem prprc1 4727
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)
Assertion
Ref Expression
prprc1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})

Proof of Theorem prprc1
StepHypRef Expression
1 snprc 4679 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 4117 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 4590 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 4114 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 4351 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2762 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2798 . 2 ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
81, 7sylbi 216 1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  Vcvv 3444  cun 3909  c0 4283  {csn 4587  {cpr 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-dif 3914  df-un 3916  df-nul 4284  df-sn 4588  df-pr 4590
This theorem is referenced by:  prprc2  4728  prprc  4729  prneprprc  4819  prex  5390  elprchashprn2  14302  elsprel  45753
  Copyright terms: Public domain W3C validator