Theorem prtlem14 38874

Description: Lemma for prter1 38879, prter2 38881 and prtex 38880. (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 38872	. . 3 ⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
2		rsp2 3255	. . 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 3933	. . . 4 ⊢ (𝑤 ∈ (𝑥 ∩ 𝑦) ↔ (𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦))
5		eq0 4316	. . . . . 6 ⊢ ((𝑥 ∩ 𝑦) = ∅ ↔ ∀𝑤 ¬ 𝑤 ∈ (𝑥 ∩ 𝑦))
6		sp 2184	. . . . . 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 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ∩ cin 3916  ∅c0 4299  Prt wprt 38871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3452  df-dif 3920  df-in 3924  df-nul 4300  df-prt 38872

This theorem is referenced by:  prtlem15  38875  prtlem17  38876