Theorem prtlem17 39375

Description: Lemma for prter2 39380. (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 3065	. . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)))
2		an32 652	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) ∧ 𝑦 ∈ 𝐴))
3		prtlem14 39373	. . . . . . . . . . 11 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)))
4		elequ2 2134	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦))
5	4	biimprd 249	. . . . . . . . . . 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 432	. . . . . . . . 9 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝑥 → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥)))))
8	7	impd 411	. . . . . . . 8 ⊢ (Prt 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑥) → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥))))
9	2, 8	biimtrrid 244	. . . . . . 7 ⊢ (Prt 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥))))
10	9	expd 416	. . . . . 6 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑦 ∈ 𝐴 → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥)))))
11	10	imp5a 441	. . . . 5 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑦 ∈ 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥))))
12	11	imp4b 422	. . . 4 ⊢ ((Prt 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)) → 𝑤 ∈ 𝑥))
13	12	exlimdv 1940	. . 3 ⊢ ((Prt 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)) → 𝑤 ∈ 𝑥))
14	1, 13	biimtrid 243	. 2 ⊢ ((Prt 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥))
15	14	ex 413	1 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1786   ∈ wcel 2119  ∃wrex 3064  Prt wprt 39370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-v 3434  df-dif 3893  df-in 3897  df-nul 4269  df-prt 39371

This theorem is referenced by:  prtlem18  39376