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Theorem prprc 4504
Description: An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
prprc ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)

Proof of Theorem prprc
StepHypRef Expression
1 prprc1 4502 . 2 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
2 snprc 4455 . . 3 𝐵 ∈ V ↔ {𝐵} = ∅)
32biimpi 207 . 2 𝐵 ∈ V → {𝐵} = ∅)
41, 3sylan9eq 2871 1 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1637  wcel 2157  Vcvv 3402  c0 4127  {csn 4381  {cpr 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-v 3404  df-dif 3783  df-un 3785  df-nul 4128  df-sn 4382  df-pr 4384
This theorem is referenced by: (None)
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