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Theorem prprc 4571
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 4569 . 2 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
2 snprc 4521 . . 3 𝐵 ∈ V ↔ {𝐵} = ∅)
32biimpi 208 . 2 𝐵 ∈ V → {𝐵} = ∅)
41, 3sylan9eq 2828 1 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1507  wcel 2048  Vcvv 3409  c0 4173  {csn 4435  {cpr 4437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411  df-dif 3828  df-un 3830  df-nul 4174  df-sn 4436  df-pr 4438
This theorem is referenced by: (None)
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