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Theorem prprc1 3545
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 3502 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 3145 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 3448 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 3142 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 3314 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2109 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2145 . 2 ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
81, 7sylbi 119 1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  wcel 1438  Vcvv 2619  cun 2995  c0 3284  {csn 3441  {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-nul 3285  df-sn 3447  df-pr 3448
This theorem is referenced by:  prprc2  3546  prprc  3547
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