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Theorem prprc1 3478
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 3435 . 2 𝐴 ∈ V ↔ {𝐴} = ∅)
2 uneq1 3090 . . 3 ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3 df-pr 3382 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 uncom 3087 . . . 4 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5 un0 3251 . . . 4 ({𝐵} ∪ ∅) = {𝐵}
64, 5eqtr2i 2061 . . 3 {𝐵} = (∅ ∪ {𝐵})
72, 3, 63eqtr4g 2097 . 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 1243  wcel 1393  Vcvv 2557  cun 2915  c0 3224  {csn 3375  {cpr 3376
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-pr 3382
This theorem is referenced by:  prprc2  3479  prprc  3480
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