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Theorem prprc1 4770
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 4722 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 4171 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 4634 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 4168 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 4400 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2764 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2800 . 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 1537  wcel 2106  Vcvv 3478  cun 3961  c0 4339  {csn 4631  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634
This theorem is referenced by:  prprc2  4771  prprc  4772  prneprprc  4866  prex  5443  prfi  9361  elprchashprn2  14432  elsprel  47400
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