Theorem prtlem17 38914

Description: Lemma for prter2 38919. (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 3057	. . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)))
2		an32 646	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) ∧ 𝑦 ∈ 𝐴))
3		prtlem14 38912	. . . . . . . . . . 11 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)))
4		elequ2 2126	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦))
5	4	biimprd 248	. . . . . . . . . . 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 243	. . . . . . 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 1934	. . 3 ⊢ ((Prt 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)) → 𝑤 ∈ 𝑥))
14	1, 13	biimtrid 242	. 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 2111  ∃wrex 3056  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-rex 3057  df-v 3438  df-dif 3905  df-in 3909  df-nul 4284  df-prt 38910

This theorem is referenced by:  prtlem18  38915