Theorem prtlem10 40015

Description: Lemma for prter3 40032. (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 490	. . . . 5 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
2		simpl 488	. . . . . 6 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → ∼ Er 𝐴)
3	2, 1	erref 8741	. . . . 5 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∼ 𝑧)
4		breq1 5105	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (𝑣 ∼ 𝑧 ↔ 𝑧 ∼ 𝑧))
5		breq1 5105	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (𝑣 ∼ 𝑤 ↔ 𝑧 ∼ 𝑤))
6	4, 5	anbi12d 644	. . . . . . 7 ⊢ (𝑣 = 𝑧 → ((𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤) ↔ (𝑧 ∼ 𝑧 ∧ 𝑧 ∼ 𝑤)))
7	6	rspcev 3576	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝑧 ∼ 𝑧 ∧ 𝑧 ∼ 𝑤)) → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
8	7	expr 462	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ∼ 𝑧) → (𝑧 ∼ 𝑤 → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
9	1, 3, 8	syl2anc 596	. . . 4 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 → ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
10		simplll 787	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → ∼ Er 𝐴)
11		simprl 783	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑧)
12		simprr 785	. . . . . 6 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑤)
13	10, 11, 12	ertr3d 8739	. . . . 5 ⊢ (((( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)) → 𝑧 ∼ 𝑤)
14	13	rexlimdva2 3165	. . . 4 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤) → 𝑧 ∼ 𝑤))
15	9, 14	impbid 215	. . 3 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤)))
16		vex 3454	. . . . . 6 ⊢ 𝑧 ∈ V
17		vex 3454	. . . . . 6 ⊢ 𝑣 ∈ V
18	16, 17	elec 8767	. . . . 5 ⊢ (𝑧 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑧)
19		vex 3454	. . . . . 6 ⊢ 𝑤 ∈ V
20	19, 17	elec 8767	. . . . 5 ⊢ (𝑤 ∈ [𝑣] ∼ ↔ 𝑣 ∼ 𝑤)
21	18, 20	anbi12i 640	. . . 4 ⊢ ((𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ) ↔ (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
22	21	rexbii 3109	. . 3 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ) ↔ ∃𝑣 ∈ 𝐴 (𝑣 ∼ 𝑧 ∧ 𝑣 ∼ 𝑤))
23	15, 22	bitr4di 292	. 2 ⊢ (( ∼ Er 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ )))
24	23	ex 418	1 ⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102   Er wer 8717  [cec 8718

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-er 8720  df-ec 8722

This theorem is used by: (None)