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Theorem prprc1 4298
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 4251 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 3758 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 4178 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 3755 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 3965 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2644 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2680 . 2 ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
81, 7sylbi 207 1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1482  wcel 1989  Vcvv 3198  cun 3570  c0 3913  {csn 4175  {cpr 4177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-dif 3575  df-un 3577  df-nul 3914  df-sn 4176  df-pr 4178
This theorem is referenced by:  prprc2  4299  prprc  4300  prex  4907  elprchashprn2  13179  elsprel  41496
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