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Theorem prprc1 3469
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 3426 . 2 A V ↔ {A} = ∅)
2 uneq1 3084 . . 3 ({A} = ∅ → ({A} ∪ {B}) = (∅ ∪ {B}))
3 df-pr 3374 . . 3 {A, B} = ({A} ∪ {B})
4 uncom 3081 . . . 4 (∅ ∪ {B}) = ({B} ∪ ∅)
5 un0 3245 . . . 4 ({B} ∪ ∅) = {B}
64, 5eqtr2i 2058 . . 3 {B} = (∅ ∪ {B})
72, 3, 63eqtr4g 2094 . 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 1242   wcel 1390  Vcvv 2551  cun 2909  c0 3218  {csn 3367  {cpr 3368
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-pr 3374
This theorem is referenced by:  prprc2  3470  prprc  3471
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