Theorem prtlem17 38210

Description: Lemma for prter2 38215. (Contributed by Rodolfo Medina, 15-Oct-2010.)

Assertion

Ref	Expression
prtlem17	⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥)))

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

Allowed substitution hints:   𝐴(𝑧,𝑤)

Proof of Theorem prtlem17

Step	Hyp	Ref	Expression
1		df-rex 3070	. . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)))
2		an32 643	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) ∧ 𝑦 ∈ 𝐴))
3		prtlem14 38208	. . . . . . . . . . 11 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)))
4		elequ2 2120	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦))
5	4	biimprd 247	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥))
6	3, 5	syl8 76	. . . . . . . . . 10 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥))))
7	6	exp4a 431	. . . . . . . . 9 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝑥 → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥)))))
8	7	impd 410	. . . . . . . 8 ⊢ (Prt 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑥) → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥))))
9	2, 8	biimtrrid 242	. . . . . . 7 ⊢ (Prt 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥))))
10	9	expd 415	. . . . . 6 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑦 ∈ 𝐴 → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥)))))
11	10	imp5a 440	. . . . 5 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑦 ∈ 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥))))
12	11	imp4b 421	. . . 4 ⊢ ((Prt 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)) → 𝑤 ∈ 𝑥))
13	12	exlimdv 1935	. . 3 ⊢ ((Prt 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)) → 𝑤 ∈ 𝑥))
14	1, 13	biimtrid 241	. 2 ⊢ ((Prt 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥))
15	14	ex 412	1 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1780   ∈ wcel 2105  ∃wrex 3069  Prt wprt 38205

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-dif 3951  df-in 3955  df-nul 4323  df-prt 38206

This theorem is referenced by:  prtlem18  38211