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Theorem prprc1 4432
 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 4385 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 3891 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 4312 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 3888 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 4098 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2771 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2807 . 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 1620   ∈ wcel 2127  Vcvv 3328   ∪ cun 3701  ∅c0 4046  {csn 4309  {cpr 4311 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-v 3330  df-dif 3706  df-un 3708  df-nul 4047  df-sn 4310  df-pr 4312 This theorem is referenced by:  prprc2  4433  prprc  4434  prneprprc  4527  prex  5046  elprchashprn2  13347  elsprel  42204
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