Theorem prtlem10 38888

Description: Lemma for prter3 38905. (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 484	. . . . 5 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
2		simpl 482	. . . . . 6 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → ∼ Er 𝐴)
3	2, 1	erref 8744	. . . . 5 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∼ 𝑧)
4		breq1 5127	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (𝑣 ∼ 𝑧 ↔ 𝑧 ∼ 𝑧))
5		breq1 5127	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (𝑣 ∼ 𝑤 ↔ 𝑧 ∼ 𝑤))
6	4, 5	anbi12d 632	. . . . . . 7 ⊢ (𝑣 = 𝑧 → ((𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤) ↔ (𝑧 ∼ 𝑧 ∧ 𝑧 ∼ 𝑤)))
7	6	rspcev 3606	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝑧 ∼ 𝑧 ∧ 𝑧 ∼ 𝑤)) → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
8	7	expr 456	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ∼ 𝑧) → (𝑧 ∼ 𝑤 → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
9	1, 3, 8	syl2anc 584	. . . 4 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
10		simplll 774	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → ∼ Er 𝐴)
11		simprl 770	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑧)
12		simprr 772	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑤)
13	10, 11, 12	ertr3d 8742	. . . . 5 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑧 ∼ 𝑤)
14	13	rexlimdva2 3144	. . . 4 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤) → 𝑧 ∼ 𝑤))
15	9, 14	impbid 212	. . 3 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
16		vex 3468	. . . . . 6 ⊢ 𝑧 ∈ V
17		vex 3468	. . . . . 6 ⊢ 𝑣 ∈ V
18	16, 17	elec 8770	. . . . 5 ⊢ (𝑧 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑧)
19		vex 3468	. . . . . 6 ⊢ 𝑤 ∈ V
20	19, 17	elec 8770	. . . . 5 ⊢ (𝑤 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑤)
21	18, 20	anbi12i 628	. . . 4 ⊢ ((𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ) ↔ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
22	21	rexbii 3084	. . 3 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ) ↔ ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
23	15, 22	bitr4di 289	. 2 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ )))
24	23	ex 412	1 ⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ∃wrex 3061   class class class wbr 5124   Er wer 8721  [cec 8722

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-er 8724  df-ec 8726

This theorem is referenced by: (None)