Theorem prtlem15 36015

Description: Lemma for prter1 36019 and prtex 36020. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Assertion

Ref	Expression
prtlem15	⊢ (Prt 𝐴 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → ∃𝑧 ∈ 𝐴 (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧)))

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

Allowed substitution hints:   𝐴(𝑤,𝑣,𝑢)

Proof of Theorem prtlem15

Step	Hyp	Ref	Expression
1		anabs7 662	. . . . . . 7 ⊢ (((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦) ∧ (𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦))) ↔ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦) ∧ (𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦)))
2		an43 656	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) ↔ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦) ∧ (𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦)))
3	2	anbi2i 624	. . . . . . 7 ⊢ (((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦))) ↔ ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦) ∧ (𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦))))
4	1, 3, 2	3bitr4ri 306	. . . . . 6 ⊢ (((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) ↔ ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦))))
5		prtlem14 36014	. . . . . . . 8 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)))
6		an3 657	. . . . . . . . 9 ⊢ (((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦))
7		elequ2 2128	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝑥 ↔ 𝑣 ∈ 𝑦))
8	7	anbi2d 630	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦)))
9	6, 8	syl5ibr 248	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥)))
10	5, 9	syl8 76	. . . . . . 7 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → (((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥)))))
11	10	imp4a 425	. . . . . 6 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦))) → (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥))))
12	4, 11	syl7bi 257	. . . . 5 ⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥))))
13	12	expdimp 455	. . . 4 ⊢ ((Prt 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐴 → (((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥))))
14	13	rexlimdv 3286	. . 3 ⊢ ((Prt 𝐴 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥)))
15	14	reximdva 3277	. 2 ⊢ (Prt 𝐴 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → ∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥)))
16		elequ2 2128	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑢 ∈ 𝑥 ↔ 𝑢 ∈ 𝑧))
17		elequ2 2128	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑣 ∈ 𝑥 ↔ 𝑣 ∈ 𝑧))
18	16, 17	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧)))
19	18	cbvrexvw 3453	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑧 ∈ 𝐴 (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧))
20	15, 19	syl6ib 253	1 ⊢ (Prt 𝐴 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → ∃𝑧 ∈ 𝐴 (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧)))

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2113  ∃wrex 3142  Prt wprt 36011

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-v 3499  df-dif 3942  df-in 3946  df-nul 4295  df-prt 36012

This theorem is referenced by:  prter1  36019