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Theorem prprc1 4693
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 4645 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 4129 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 4560 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 4126 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 4341 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2842 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2878 . 2 ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
81, 7sylbi 218 1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  c0 4288  {csn 4557  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-un 3938  df-nul 4289  df-sn 4558  df-pr 4560
This theorem is referenced by:  prprc2  4694  prprc  4695  prneprprc  4783  prex  5323  elprchashprn2  13745  elsprel  43514
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