Theorem prtlem400 36447

Description: Lemma for prter2 36458 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 2961	. 2 ⊢ ¬ ∅ ≠ ∅
2		prtlem13.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
3	2	prtlem16 36446	. . 3 ⊢ dom ∼ = ∪ 𝐴
4		elqsn0 8377	. . 3 ⊢ ((dom ∼ = ∪ 𝐴 ∧ ∅ ∈ (∪ 𝐴 / ∼ )) → ∅ ≠ ∅)
5	3, 4	mpan 690	. 2 ⊢ (∅ ∈ (∪ 𝐴 / ∼ ) → ∅ ≠ ∅)
6	1, 5	mto 200	1 ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ )

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∃wrex 3072  ∅c0 4226  ∪ cuni 4799  {copab 5095  dom cdm 5525   / cqs 8299

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ec 8302  df-qs 8306

This theorem is referenced by:  prter2  36458