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Theorem frgrwopreglem3 41485
Description: Lemma 3 for frgrwopreg 41488. The vertices in the sets A and B have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrwopreglem3 ((𝑋𝐴𝑌𝐵) → (𝐷𝑋) ≠ (𝐷𝑌))
Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem frgrwopreglem3
StepHypRef Expression
1 fveq2 6088 . . . . 5 (𝑥 = 𝑋 → (𝐷𝑥) = (𝐷𝑋))
21eqeq1d 2611 . . . 4 (𝑥 = 𝑋 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑋) = 𝐾))
3 frgrwopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
42, 3elrab2 3332 . . 3 (𝑋𝐴 ↔ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾))
5 frgrwopreg.b . . . . . 6 𝐵 = (𝑉𝐴)
65eleq2i 2679 . . . . 5 (𝑌𝐵𝑌 ∈ (𝑉𝐴))
7 eldif 3549 . . . . 5 (𝑌 ∈ (𝑉𝐴) ↔ (𝑌𝑉 ∧ ¬ 𝑌𝐴))
86, 7bitri 262 . . . 4 (𝑌𝐵 ↔ (𝑌𝑉 ∧ ¬ 𝑌𝐴))
9 fveq2 6088 . . . . . . . . 9 (𝑥 = 𝑌 → (𝐷𝑥) = (𝐷𝑌))
109eqeq1d 2611 . . . . . . . 8 (𝑥 = 𝑌 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑌) = 𝐾))
1110, 3elrab2 3332 . . . . . . 7 (𝑌𝐴 ↔ (𝑌𝑉 ∧ (𝐷𝑌) = 𝐾))
12 ianor 507 . . . . . . . 8 (¬ (𝑌𝑉 ∧ (𝐷𝑌) = 𝐾) ↔ (¬ 𝑌𝑉 ∨ ¬ (𝐷𝑌) = 𝐾))
13 pm2.21 118 . . . . . . . . 9 𝑌𝑉 → (𝑌𝑉 → ((𝑋𝑉 ∧ (𝐷𝑋) = 𝐾) → (𝐷𝑋) ≠ (𝐷𝑌))))
14 neqne 2789 . . . . . . . . . . . . . 14 (¬ (𝐷𝑌) = 𝐾 → (𝐷𝑌) ≠ 𝐾)
1514adantr 479 . . . . . . . . . . . . 13 ((¬ (𝐷𝑌) = 𝐾 ∧ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾)) → (𝐷𝑌) ≠ 𝐾)
1615necomd 2836 . . . . . . . . . . . 12 ((¬ (𝐷𝑌) = 𝐾 ∧ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾)) → 𝐾 ≠ (𝐷𝑌))
17 neeq1 2843 . . . . . . . . . . . . 13 ((𝐷𝑋) = 𝐾 → ((𝐷𝑋) ≠ (𝐷𝑌) ↔ 𝐾 ≠ (𝐷𝑌)))
1817ad2antll 760 . . . . . . . . . . . 12 ((¬ (𝐷𝑌) = 𝐾 ∧ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾)) → ((𝐷𝑋) ≠ (𝐷𝑌) ↔ 𝐾 ≠ (𝐷𝑌)))
1916, 18mpbird 245 . . . . . . . . . . 11 ((¬ (𝐷𝑌) = 𝐾 ∧ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾)) → (𝐷𝑋) ≠ (𝐷𝑌))
2019ex 448 . . . . . . . . . 10 (¬ (𝐷𝑌) = 𝐾 → ((𝑋𝑉 ∧ (𝐷𝑋) = 𝐾) → (𝐷𝑋) ≠ (𝐷𝑌)))
2120a1d 25 . . . . . . . . 9 (¬ (𝐷𝑌) = 𝐾 → (𝑌𝑉 → ((𝑋𝑉 ∧ (𝐷𝑋) = 𝐾) → (𝐷𝑋) ≠ (𝐷𝑌))))
2213, 21jaoi 392 . . . . . . . 8 ((¬ 𝑌𝑉 ∨ ¬ (𝐷𝑌) = 𝐾) → (𝑌𝑉 → ((𝑋𝑉 ∧ (𝐷𝑋) = 𝐾) → (𝐷𝑋) ≠ (𝐷𝑌))))
2312, 22sylbi 205 . . . . . . 7 (¬ (𝑌𝑉 ∧ (𝐷𝑌) = 𝐾) → (𝑌𝑉 → ((𝑋𝑉 ∧ (𝐷𝑋) = 𝐾) → (𝐷𝑋) ≠ (𝐷𝑌))))
2411, 23sylnbi 318 . . . . . 6 𝑌𝐴 → (𝑌𝑉 → ((𝑋𝑉 ∧ (𝐷𝑋) = 𝐾) → (𝐷𝑋) ≠ (𝐷𝑌))))
2524impcom 444 . . . . 5 ((𝑌𝑉 ∧ ¬ 𝑌𝐴) → ((𝑋𝑉 ∧ (𝐷𝑋) = 𝐾) → (𝐷𝑋) ≠ (𝐷𝑌)))
2625com12 32 . . . 4 ((𝑋𝑉 ∧ (𝐷𝑋) = 𝐾) → ((𝑌𝑉 ∧ ¬ 𝑌𝐴) → (𝐷𝑋) ≠ (𝐷𝑌)))
278, 26syl5bi 230 . . 3 ((𝑋𝑉 ∧ (𝐷𝑋) = 𝐾) → (𝑌𝐵 → (𝐷𝑋) ≠ (𝐷𝑌)))
284, 27sylbi 205 . 2 (𝑋𝐴 → (𝑌𝐵 → (𝐷𝑋) ≠ (𝐷𝑌)))
2928imp 443 1 ((𝑋𝐴𝑌𝐵) → (𝐷𝑋) ≠ (𝐷𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  {crab 2899  cdif 3536  cfv 5790  Vtxcvtx 40231  VtxDegcvtxdg 40683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798
This theorem is referenced by:  frgrwopreglem4  41486
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