Theorem prtlem19 39652

Description: Lemma for prter2 39655. (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 39651	. . . . 5 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤)))
3	2	imp 411	. . . 4 ⊢ ((Prt 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤))
4		vex 3459	. . . . 5 ⊢ 𝑤 ∈ V
5		vex 3459	. . . . 5 ⊢ 𝑧 ∈ V
6	4, 5	elec 8737	. . . 4 ⊢ (𝑤 ∈ [𝑧] ∼ ↔ 𝑧 ∼ 𝑤)
7	3, 6	bitr4di 292	. . 3 ⊢ ((Prt 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ [𝑧] ∼ ))
8	7	eqrdv 2761	. 2 ⊢ ((Prt 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 = [𝑧] ∼ )
9	8	ex 417	1 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   class class class wbr 5109  {copab 5173  [cec 8688  Prt wprt 39645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-prt 39646

This theorem is referenced by:  prter2  39655