Theorem prtlem14 40024

Description: Lemma for prter1 40029, prter2 40031 and prtex 40030. (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 40022	. . 3 ⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
2		rsp2 3279	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)))
3	1, 2	sylbi 220	. 2 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)))
4		elin 4093	. . . 4 ⊢ (𝑤 ∈ (𝑥 ∩ 𝑦) ↔ (𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦))
5		eq0 4296	. . . . . 6 ⊢ ((𝑥 ∩ 𝑦) = ∅ ↔ ∀𝑤 ¬ 𝑤 ∈ (𝑥 ∩ 𝑦))
6		sp 2219	. . . . . 6 ⊢ (∀𝑤 ¬ 𝑤 ∈ (𝑥 ∩ 𝑦) → ¬ 𝑤 ∈ (𝑥 ∩ 𝑦))
7	5, 6	sylbi 220	. . . . 5 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ¬ 𝑤 ∈ (𝑥 ∩ 𝑦))
8	7	pm2.21d 122	. . . 4 ⊢ ((𝑥 ∩ 𝑦) = ∅ → (𝑤 ∈ (𝑥 ∩ 𝑦) → 𝑥 = 𝑦))
9	4, 8	biimtrrid 246	. . 3 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦))
10	9	jao1i 872	. 2 ⊢ ((𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦))
11	3, 10	syl6 36	1 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∩ cin 4089  ∅c0 4278  Prt wprt 40021

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-dif 4047  df-in 4091  df-nul 4279  df-prt 40022

This theorem is used by:  prtlem15  40025  prtlem17  40026