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Theorem prprc1 3452
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
prprc1 A V → {A, B} = {B})

Proof of Theorem prprc1
StepHypRef Expression
1 snprc 3409 . 2 A V ↔ {A} = ∅)
2 uneq1 3067 . . 3 ({A} = ∅ → ({A} ∪ {B}) = (∅ ∪ {B}))
3 df-pr 3357 . . 3 {A, B} = ({A} ∪ {B})
4 uncom 3064 . . . 4 (∅ ∪ {B}) = ({B} ∪ ∅)
5 un0 3228 . . . 4 ({B} ∪ ∅) = {B}
64, 5eqtr2i 2043 . . 3 {B} = (∅ ∪ {B})
72, 3, 63eqtr4g 2079 . 2 ({A} = ∅ → {A, B} = {B})
81, 7sylbi 114 1 A V → {A, B} = {B})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1228   wcel 1374  Vcvv 2535  cun 2892  c0 3201  {csn 3350  {cpr 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-pr 3357
This theorem is referenced by:  prprc2  3453  prprc  3454
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