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Theorem prprc2 4122
 Description: An unordered pair of a proper class. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
prprc2 A V → {B, A} = {B})

Proof of Theorem prprc2
StepHypRef Expression
1 df-pr 3742 . 2 {B, A} = ({B} ∪ {A})
2 snprc 3788 . . . . 5 A V ↔ {A} = )
32biimpi 186 . . . 4 A V → {A} = )
43uneq2d 3418 . . 3 A V → ({B} ∪ {A}) = ({B} ∪ ))
5 un0 3575 . . 3 ({B} ∪ ) = {B}
64, 5syl6eq 2401 . 2 A V → ({B} ∪ {A}) = {B})
71, 6syl5eq 2397 1 A V → {B, A} = {B})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207  ∅c0 3550  {csn 3737  {cpr 3738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-nul 3551  df-sn 3741  df-pr 3742 This theorem is referenced by:  prprc1  4123
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