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Theorem prprc1 4765
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 4717 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 4161 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 4629 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 4158 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 4394 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2766 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2802 . 2 ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
81, 7sylbi 217 1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  c0 4333  {csn 4626  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629
This theorem is referenced by:  prprc2  4766  prprc  4767  prneprprc  4861  prex  5437  prfi  9363  elprchashprn2  14435  elsprel  47462
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