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Theorem prprc1 4243
 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 4197 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 3722 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 4128 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 3719 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 3919 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2633 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2669 . 2 ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
81, 7sylbi 206 1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128 This theorem is referenced by:  prprc2  4244  prprc  4245  prex  4836  elprchashprn2  13045  usgraedgprv  25905
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