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Theorem prtlem14 35877
Description: Lemma for prter1 35882, prter2 35884 and prtex 35883. (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 35875 . . 3 (Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
2 rsp2 3217 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)))
31, 2sylbi 218 . 2 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)))
4 elin 4172 . . . 4 (𝑤 ∈ (𝑥𝑦) ↔ (𝑤𝑥𝑤𝑦))
5 eq0 4311 . . . . . 6 ((𝑥𝑦) = ∅ ↔ ∀𝑤 ¬ 𝑤 ∈ (𝑥𝑦))
6 sp 2174 . . . . . 6 (∀𝑤 ¬ 𝑤 ∈ (𝑥𝑦) → ¬ 𝑤 ∈ (𝑥𝑦))
75, 6sylbi 218 . . . . 5 ((𝑥𝑦) = ∅ → ¬ 𝑤 ∈ (𝑥𝑦))
87pm2.21d 121 . . . 4 ((𝑥𝑦) = ∅ → (𝑤 ∈ (𝑥𝑦) → 𝑥 = 𝑦))
94, 8syl5bir 244 . . 3 ((𝑥𝑦) = ∅ → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦))
109jao1i 854 . 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 396  wo 843  wal 1528   = wceq 1530  wcel 2107  wral 3142  cin 3938  c0 4294  Prt wprt 35874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-v 3501  df-dif 3942  df-in 3946  df-nul 4295  df-prt 35875
This theorem is referenced by:  prtlem15  35878  prtlem17  35879
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