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Theorem frgrawopreglem3 26573
 Description: Lemma 3 for frgrawopreg 26576. The vertices in the sets A and B have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
Hypotheses
Ref Expression
frgrawopreg.a 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrawopreglem3 ((𝑋𝐴𝑌𝐵) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem frgrawopreglem3
StepHypRef Expression
1 fveq2 6103 . . . . 5 (𝑥 = 𝑋 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑋))
21eqeq1d 2612 . . . 4 (𝑥 = 𝑋 → (((𝑉 VDeg 𝐸)‘𝑥) = 𝐾 ↔ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾))
3 frgrawopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
42, 3elrab2 3333 . . 3 (𝑋𝐴 ↔ (𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾))
5 frgrawopreg.b . . . . . 6 𝐵 = (𝑉𝐴)
65eleq2i 2680 . . . . 5 (𝑌𝐵𝑌 ∈ (𝑉𝐴))
7 eldif 3550 . . . . 5 (𝑌 ∈ (𝑉𝐴) ↔ (𝑌𝑉 ∧ ¬ 𝑌𝐴))
86, 7bitri 263 . . . 4 (𝑌𝐵 ↔ (𝑌𝑉 ∧ ¬ 𝑌𝐴))
9 fveq2 6103 . . . . . . . . 9 (𝑥 = 𝑌 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑌))
109eqeq1d 2612 . . . . . . . 8 (𝑥 = 𝑌 → (((𝑉 VDeg 𝐸)‘𝑥) = 𝐾 ↔ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾))
1110, 3elrab2 3333 . . . . . . 7 (𝑌𝐴 ↔ (𝑌𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾))
12 ianor 508 . . . . . . . 8 (¬ (𝑌𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾) ↔ (¬ 𝑌𝑉 ∨ ¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾))
13 pm2.21 119 . . . . . . . . 9 𝑌𝑉 → (𝑌𝑉 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
14 nesym 2838 . . . . . . . . . . . . . 14 (𝐾 ≠ ((𝑉 VDeg 𝐸)‘𝑌) ↔ ¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾)
1514biimpri 217 . . . . . . . . . . . . 13 (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾𝐾 ≠ ((𝑉 VDeg 𝐸)‘𝑌))
16 neeq1 2844 . . . . . . . . . . . . 13 (((𝑉 VDeg 𝐸)‘𝑋) = 𝐾 → (((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌) ↔ 𝐾 ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
1715, 16syl5ibr 235 . . . . . . . . . . . 12 (((𝑉 VDeg 𝐸)‘𝑋) = 𝐾 → (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
1817adantl 481 . . . . . . . . . . 11 ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
1918com12 32 . . . . . . . . . 10 (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
2019a1d 25 . . . . . . . . 9 (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → (𝑌𝑉 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
2113, 20jaoi 393 . . . . . . . 8 ((¬ 𝑌𝑉 ∨ ¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾) → (𝑌𝑉 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
2212, 21sylbi 206 . . . . . . 7 (¬ (𝑌𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾) → (𝑌𝑉 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
2311, 22sylnbi 319 . . . . . 6 𝑌𝐴 → (𝑌𝑉 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
2423impcom 445 . . . . 5 ((𝑌𝑉 ∧ ¬ 𝑌𝐴) → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
2524com12 32 . . . 4 ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑌𝑉 ∧ ¬ 𝑌𝐴) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
268, 25syl5bi 231 . . 3 ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → (𝑌𝐵 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
274, 26sylbi 206 . 2 (𝑋𝐴 → (𝑌𝐵 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
2827imp 444 1 ((𝑋𝐴𝑌𝐵) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∖ cdif 3537  ‘cfv 5804  (class class class)co 6549   VDeg cvdg 26420 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by:  frgrawopreglem4  26574
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