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Theorem prtlem400 37382
Description: Lemma for prter2 37393 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 37381 . . 3 dom = 𝐴
4 elqsn0 8731 . . 3 ((dom = 𝐴 ∧ ∅ ∈ ( 𝐴 / )) → ∅ ≠ ∅)
53, 4mpan 689 . 2 (∅ ∈ ( 𝐴 / ) → ∅ ≠ ∅)
61, 5mto 196 1 ¬ ∅ ∈ ( 𝐴 / )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1542  wcel 2107  wne 2940  wrex 3070  c0 4286   cuni 4869  {copab 5171  dom cdm 5637   / cqs 8653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ec 8656  df-qs 8660
This theorem is referenced by:  prter2  37393
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