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Theorem prtlem400 38871
Description: Lemma for prter2 38882 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem13.1 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Assertion
Ref Expression
prtlem400 ¬ ∅ ∈ ( 𝐴 / )
Distinct variable group:   𝑥,𝑢,𝑦,𝐴
Allowed substitution hints:   (𝑥,𝑦,𝑢)

Proof of Theorem prtlem400
StepHypRef Expression
1 neirr 2949 . 2 ¬ ∅ ≠ ∅
2 prtlem13.1 . . . 4 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
32prtlem16 38870 . . 3 dom = 𝐴
4 elqsn0 8826 . . 3 ((dom = 𝐴 ∧ ∅ ∈ ( 𝐴 / )) → ∅ ≠ ∅)
53, 4mpan 690 . 2 (∅ ∈ ( 𝐴 / ) → ∅ ≠ ∅)
61, 5mto 197 1 ¬ ∅ ∈ ( 𝐴 / )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wcel 2108  wne 2940  wrex 3070  c0 4333   cuni 4907  {copab 5205  dom cdm 5685   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751
This theorem is referenced by:  prter2  38882
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