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Theorem prtlem14 34895
Description: Lemma for prter1 34900, prter2 34902 and prtex 34901. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
prtlem14 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑤,𝑦   𝑥,𝐴,𝑦
Allowed substitution hint:   𝐴(𝑤)

Proof of Theorem prtlem14
StepHypRef Expression
1 df-prt 34893 . . 3 (Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
2 rsp2 3117 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)))
31, 2sylbi 209 . 2 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)))
4 elin 3994 . . . 4 (𝑤 ∈ (𝑥𝑦) ↔ (𝑤𝑥𝑤𝑦))
5 eq0 4129 . . . . . 6 ((𝑥𝑦) = ∅ ↔ ∀𝑤 ¬ 𝑤 ∈ (𝑥𝑦))
6 sp 2217 . . . . . 6 (∀𝑤 ¬ 𝑤 ∈ (𝑥𝑦) → ¬ 𝑤 ∈ (𝑥𝑦))
75, 6sylbi 209 . . . . 5 ((𝑥𝑦) = ∅ → ¬ 𝑤 ∈ (𝑥𝑦))
87pm2.21d 119 . . . 4 ((𝑥𝑦) = ∅ → (𝑤 ∈ (𝑥𝑦) → 𝑥 = 𝑦))
94, 8syl5bir 235 . . 3 ((𝑥𝑦) = ∅ → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦))
109jao1i 885 . 2 ((𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦))
113, 10syl6 35 1 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874  wal 1651   = wceq 1653  wcel 2157  wral 3089  cin 3768  c0 4115  Prt wprt 34892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-dif 3772  df-in 3776  df-nul 4116  df-prt 34893
This theorem is referenced by:  prtlem15  34896  prtlem17  34897
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