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Theorem prtlem14 37386
Description: Lemma for prter1 37391, prter2 37393 and prtex 37392. (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 37384 . . 3 (Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
2 rsp2 3259 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)))
31, 2sylbi 216 . 2 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)))
4 elin 3930 . . . 4 (𝑤 ∈ (𝑥𝑦) ↔ (𝑤𝑥𝑤𝑦))
5 eq0 4307 . . . . . 6 ((𝑥𝑦) = ∅ ↔ ∀𝑤 ¬ 𝑤 ∈ (𝑥𝑦))
6 sp 2177 . . . . . 6 (∀𝑤 ¬ 𝑤 ∈ (𝑥𝑦) → ¬ 𝑤 ∈ (𝑥𝑦))
75, 6sylbi 216 . . . . 5 ((𝑥𝑦) = ∅ → ¬ 𝑤 ∈ (𝑥𝑦))
87pm2.21d 121 . . . 4 ((𝑥𝑦) = ∅ → (𝑤 ∈ (𝑥𝑦) → 𝑥 = 𝑦))
94, 8biimtrrid 242 . . 3 ((𝑥𝑦) = ∅ → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦))
109jao1i 857 . 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 397  wo 846  wal 1540   = wceq 1542  wcel 2107  wral 3061  cin 3913  c0 4286  Prt wprt 37383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3449  df-dif 3917  df-in 3921  df-nul 4287  df-prt 37384
This theorem is referenced by:  prtlem15  37387  prtlem17  37388
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