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Theorem prprc1 3601
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 3558 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 3193 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 3504 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 3190 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 3366 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2139 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2175 . 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 1316  wcel 1465  Vcvv 2660  cun 3039  c0 3333  {csn 3497  {cpr 3498
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-un 3045  df-nul 3334  df-sn 3503  df-pr 3504
This theorem is referenced by:  prprc2  3602  prprc  3603
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