Theorem prtlem10 35013

Description: Lemma for prter3 35030. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
prtlem10	⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ))))

Distinct variable groups:   𝑤,𝑣   𝑧,𝑣   𝑣,𝐴   𝑣, ∼

Allowed substitution hints:   𝐴(𝑧,𝑤)   ∼ (𝑧,𝑤)

Proof of Theorem prtlem10

Step	Hyp	Ref	Expression
1		simpr 479	. . . . 5 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
2		simpl 476	. . . . . 6 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → ∼ Er 𝐴)
3	2, 1	erref 8046	. . . . 5 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∼ 𝑧)
4		breq1 4889	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (𝑣 ∼ 𝑧 ↔ 𝑧 ∼ 𝑧))
5		breq1 4889	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (𝑣 ∼ 𝑤 ↔ 𝑧 ∼ 𝑤))
6	4, 5	anbi12d 624	. . . . . . 7 ⊢ (𝑣 = 𝑧 → ((𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤) ↔ (𝑧 ∼ 𝑧 ∧ 𝑧 ∼ 𝑤)))
7	6	rspcev 3510	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝑧 ∼ 𝑧 ∧ 𝑧 ∼ 𝑤)) → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
8	7	expr 450	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ∼ 𝑧) → (𝑧 ∼ 𝑤 → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
9	1, 3, 8	syl2anc 579	. . . 4 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
10		simplll 765	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → ∼ Er 𝐴)
11		simprl 761	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑧)
12		simprr 763	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑤)
13	10, 11, 12	ertr3d 8044	. . . . 5 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑧 ∼ 𝑤)
14	13	rexlimdva2 3215	. . . 4 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤) → 𝑧 ∼ 𝑤))
15	9, 14	impbid 204	. . 3 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
16		vex 3400	. . . . . 6 ⊢ 𝑧 ∈ V
17		vex 3400	. . . . . 6 ⊢ 𝑣 ∈ V
18	16, 17	elec 8068	. . . . 5 ⊢ (𝑧 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑧)
19		vex 3400	. . . . . 6 ⊢ 𝑤 ∈ V
20	19, 17	elec 8068	. . . . 5 ⊢ (𝑤 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑤)
21	18, 20	anbi12i 620	. . . 4 ⊢ ((𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ) ↔ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
22	21	rexbii 3223	. . 3 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ) ↔ ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
23	15, 22	syl6bbr 281	. 2 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ )))
24	23	ex 403	1 ⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2106  ∃wrex 3090   class class class wbr 4886   Er wer 8023  [cec 8024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-er 8026  df-ec 8028

This theorem is referenced by: (None)