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Theorem prprc1 3506
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 3463 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 3118 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 3410 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 3115 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 3279 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2077 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2113 . 2 ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
81, 7sylbi 118 1 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1259  wcel 1409  Vcvv 2574  cun 2943  c0 3252  {csn 3403  {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950  df-nul 3253  df-sn 3409  df-pr 3410
This theorem is referenced by:  prprc2  3507  prprc  3508
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