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Theorem prprc1 4686
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 4638 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 4118 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 4553 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 4115 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 4327 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2848 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2884 . 2 ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
81, 7sylbi 220 1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  cun 3917  c0 4276  {csn 4550  {cpr 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-nul 4277  df-sn 4551  df-pr 4553
This theorem is referenced by:  prprc2  4687  prprc  4688  prneprprc  4775  prex  5320  elprchashprn2  13762  elsprel  43918
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