Theorem prtlem14 38912

Description: Lemma for prter1 38917, prter2 38919 and prtex 38918. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Assertion

Ref	Expression
prtlem14	⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)))

Distinct variable groups:   𝑥,𝑤,𝑦   𝑥,𝐴,𝑦

Allowed substitution hint:   𝐴(𝑤)

Proof of Theorem prtlem14

Step	Hyp	Ref	Expression
1		df-prt 38910	. . 3 ⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
2		rsp2 3249	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)))
3	1, 2	sylbi 217	. 2 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)))
4		elin 3918	. . . 4 ⊢ (𝑤 ∈ (𝑥 ∩ 𝑦) ↔ (𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦))
5		eq0 4300	. . . . . 6 ⊢ ((𝑥 ∩ 𝑦) = ∅ ↔ ∀𝑤 ¬ 𝑤 ∈ (𝑥 ∩ 𝑦))
6		sp 2186	. . . . . 6 ⊢ (∀𝑤 ¬ 𝑤 ∈ (𝑥 ∩ 𝑦) → ¬ 𝑤 ∈ (𝑥 ∩ 𝑦))
7	5, 6	sylbi 217	. . . . 5 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ¬ 𝑤 ∈ (𝑥 ∩ 𝑦))
8	7	pm2.21d 121	. . . 4 ⊢ ((𝑥 ∩ 𝑦) = ∅ → (𝑤 ∈ (𝑥 ∩ 𝑦) → 𝑥 = 𝑦))
9	4, 8	biimtrrid 243	. . 3 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦))
10	9	jao1i 858	. 2 ⊢ ((𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦))
11	3, 10	syl6 35	1 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∩ cin 3901  ∅c0 4283  Prt wprt 38909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-dif 3905  df-in 3909  df-nul 4284  df-prt 38910

This theorem is referenced by:  prtlem15  38913  prtlem17  38914