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Theorem prprc1 3635
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 3592 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 3224 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 3535 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 3221 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 3397 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2162 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2198 . 2 ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
81, 7sylbi 120 1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1332  wcel 1481  Vcvv 2687  cun 3070  c0 3364  {csn 3528  {cpr 3529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-dif 3074  df-un 3076  df-nul 3365  df-sn 3534  df-pr 3535
This theorem is referenced by:  prprc2  3636  prprc  3637
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