Theorem prtlem10 37400

Description: Lemma for prter3 37417. (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 485	. . . . 5 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
2		simpl 483	. . . . . 6 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → ∼ Er 𝐴)
3	2, 1	erref 8675	. . . . 5 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∼ 𝑧)
4		breq1 5113	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (𝑣 ∼ 𝑧 ↔ 𝑧 ∼ 𝑧))
5		breq1 5113	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (𝑣 ∼ 𝑤 ↔ 𝑧 ∼ 𝑤))
6	4, 5	anbi12d 631	. . . . . . 7 ⊢ (𝑣 = 𝑧 → ((𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤) ↔ (𝑧 ∼ 𝑧 ∧ 𝑧 ∼ 𝑤)))
7	6	rspcev 3582	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝑧 ∼ 𝑧 ∧ 𝑧 ∼ 𝑤)) → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
8	7	expr 457	. . . . 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 773	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → ∼ Er 𝐴)
11		simprl 769	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑧)
12		simprr 771	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑤)
13	10, 11, 12	ertr3d 8673	. . . . 5 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑧 ∼ 𝑤)
14	13	rexlimdva2 3150	. . . 4 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤) → 𝑧 ∼ 𝑤))
15	9, 14	impbid 211	. . 3 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
16		vex 3450	. . . . . 6 ⊢ 𝑧 ∈ V
17		vex 3450	. . . . . 6 ⊢ 𝑣 ∈ V
18	16, 17	elec 8699	. . . . 5 ⊢ (𝑧 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑧)
19		vex 3450	. . . . . 6 ⊢ 𝑤 ∈ V
20	19, 17	elec 8699	. . . . 5 ⊢ (𝑤 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑤)
21	18, 20	anbi12i 627	. . . 4 ⊢ ((𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ) ↔ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
22	21	rexbii 3093	. . 3 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ) ↔ ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
23	15, 22	bitr4di 288	. 2 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ )))
24	23	ex 413	1 ⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∃wrex 3069   class class class wbr 5110   Er wer 8652  [cec 8653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-er 8655  df-ec 8657

This theorem is referenced by: (None)