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Theorem frgrawopreglem3 26339
Description: Lemma 3 for frgrawopreg 26342. 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 6088 . . . . 5 (𝑥 = 𝑋 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑋))
21eqeq1d 2611 . . . 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 3332 . . 3 (𝑋𝐴 ↔ (𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾))
5 frgrawopreg.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 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑌))
109eqeq1d 2611 . . . . . . . 8 (𝑥 = 𝑌 → (((𝑉 VDeg 𝐸)‘𝑥) = 𝐾 ↔ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾))
1110, 3elrab2 3332 . . . . . . 7 (𝑌𝐴 ↔ (𝑌𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾))
12 ianor 507 . . . . . . . 8 (¬ (𝑌𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾) ↔ (¬ 𝑌𝑉 ∨ ¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾))
13 pm2.21 118 . . . . . . . . 9 𝑌𝑉 → (𝑌𝑉 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
14 nesym 2837 . . . . . . . . . . . . . 14 (𝐾 ≠ ((𝑉 VDeg 𝐸)‘𝑌) ↔ ¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾)
1514biimpri 216 . . . . . . . . . . . . 13 (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾𝐾 ≠ ((𝑉 VDeg 𝐸)‘𝑌))
16 neeq1 2843 . . . . . . . . . . . . 13 (((𝑉 VDeg 𝐸)‘𝑋) = 𝐾 → (((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌) ↔ 𝐾 ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
1715, 16syl5ibr 234 . . . . . . . . . . . 12 (((𝑉 VDeg 𝐸)‘𝑋) = 𝐾 → (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
1817adantl 480 . . . . . . . . . . 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 392 . . . . . . . 8 ((¬ 𝑌𝑉 ∨ ¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾) → (𝑌𝑉 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
2212, 21sylbi 205 . . . . . . 7 (¬ (𝑌𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾) → (𝑌𝑉 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
2311, 22sylnbi 318 . . . . . 6 𝑌𝐴 → (𝑌𝑉 → ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
2423impcom 444 . . . . 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 230 . . 3 ((𝑋𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → (𝑌𝐵 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
274, 26sylbi 205 . 2 (𝑋𝐴 → (𝑌𝐵 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
2827imp 443 1 ((𝑋𝐴𝑌𝐵) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  {crab 2899  cdif 3536  cfv 5790  (class class class)co 6527   VDeg cvdg 26186
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:  frgrawopreglem4  26340
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