Theorem prtlem19 39138

Description: Lemma for prter2 39141. (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 39137	. . . . 5 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤)))
3	2	imp 406	. . . 4 ⊢ ((Prt 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤))
4		vex 3444	. . . . 5 ⊢ 𝑤 ∈ V
5		vex 3444	. . . . 5 ⊢ 𝑧 ∈ V
6	4, 5	elec 8681	. . . 4 ⊢ (𝑤 ∈ [𝑧] ∼ ↔ 𝑧 ∼ 𝑤)
7	3, 6	bitr4di 289	. . 3 ⊢ ((Prt 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ [𝑧] ∼ ))
8	7	eqrdv 2734	. 2 ⊢ ((Prt 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 = [𝑧] ∼ )
9	8	ex 412	1 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   class class class wbr 5098  {copab 5160  [cec 8633  Prt wprt 39131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637  df-prt 39132

This theorem is referenced by:  prter2  39141