Theorem prtlem10 37821

Description: Lemma for prter3 37838. (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 8725	. . . . 5 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∼ 𝑧)
4		breq1 5151	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (𝑣 ∼ 𝑧 ↔ 𝑧 ∼ 𝑧))
5		breq1 5151	. . . . . . . 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 3612	. . . . . 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 8723	. . . . 5 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑧 ∼ 𝑤)
14	13	rexlimdva2 3157	. . . 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 3478	. . . . . 6 ⊢ 𝑧 ∈ V
17		vex 3478	. . . . . 6 ⊢ 𝑣 ∈ V
18	16, 17	elec 8749	. . . . 5 ⊢ (𝑧 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑧)
19		vex 3478	. . . . . 6 ⊢ 𝑤 ∈ V
20	19, 17	elec 8749	. . . . 5 ⊢ (𝑤 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑤)
21	18, 20	anbi12i 627	. . . 4 ⊢ ((𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ) ↔ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
22	21	rexbii 3094	. . 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 3070   class class class wbr 5148   Er wer 8702  [cec 8703

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-er 8705  df-ec 8707

This theorem is referenced by: (None)