Theorem prtlem400 39240

Description: Lemma for prter2 39251 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⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}

Assertion

Ref	Expression
prtlem400	⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ )

Distinct variable group:   𝑥,𝑢,𝑦,𝐴

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑢)

Proof of Theorem prtlem400

Step	Hyp	Ref	Expression
1		neirr 2942	. 2 ⊢ ¬ ∅ ≠ ∅
2		prtlem13.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
3	2	prtlem16 39239	. . 3 ⊢ dom ∼ = ∪ 𝐴
4		elqsn0 8733	. . 3 ⊢ ((dom ∼ = ∪ 𝐴 ∧ ∅ ∈ (∪ 𝐴 / ∼ )) → ∅ ≠ ∅)
5	3, 4	mpan 691	. 2 ⊢ (∅ ∈ (∪ 𝐴 / ∼ ) → ∅ ≠ ∅)
6	1, 5	mto 197	1 ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ )

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  ∅c0 4287  ∪ cuni 4865  {copab 5162  dom cdm 5632   / cqs 8644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8647  df-qs 8651

This theorem is referenced by:  prter2  39251