Theorem prtlem19 37200

Description: Lemma for prter2 37203. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem18.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}

Assertion

Ref	Expression
prtlem19	⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))

Distinct variable groups:   𝑣,𝑢,𝑥,𝑦,𝑧,𝐴   𝑣, ∼ ,𝑧

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑢)

Proof of Theorem prtlem19

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prtlem18.1	. . . . . 6 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
2	1	prtlem18 37199	. . . . 5 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤)))
3	2	imp 408	. . . 4 ⊢ ((Prt 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤))
4		vex 3447	. . . . 5 ⊢ 𝑤 ∈ V
5		vex 3447	. . . . 5 ⊢ 𝑧 ∈ V
6	4, 5	elec 8625	. . . 4 ⊢ (𝑤 ∈ [𝑧] ∼ ↔ 𝑧 ∼ 𝑤)
7	3, 6	bitr4di 289	. . 3 ⊢ ((Prt 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ [𝑧] ∼ ))
8	7	eqrdv 2735	. 2 ⊢ ((Prt 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 = [𝑧] ∼ )
9	8	ex 414	1 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1541   ∈ wcel 2106  ∃wrex 3071   class class class wbr 5103  {copab 5165  [cec 8579  Prt wprt 37193

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-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5636  df-cnv 5638  df-dm 5640  df-rn 5641  df-res 5642  df-ima 5643  df-ec 8583  df-prt 37194

This theorem is referenced by:  prter2  37203