Theorem prtlem400 39895

Description: Lemma for prter2 39906 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 2965	. 2 ⊢ ¬ ∅ ≠ ∅
2		prtlem13.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
3	2	prtlem16 39894	. . 3 ⊢ dom ∼ = ∪ 𝐴
4		elqsn0 8789	. . 3 ⊢ ((dom ∼ = ∪ 𝐴 ∧ ∅ ∈ (∪ 𝐴 / ∼ )) → ∅ ≠ ∅)
5	3, 4	mpan 703	. 2 ⊢ (∅ ∈ (∪ 𝐴 / ∼ ) → ∅ ≠ ∅)
6	1, 5	mto 200	1 ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ )

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  ∅c0 4279  ∪ cuni 4867  {copab 5167  dom cdm 5651   / cqs 8700

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8703  df-qs 8707

This theorem is used by:  prter2  39906