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Theorem kur14lem1 30442
 Description: Lemma for kur14 30452. (Contributed by Mario Carneiro, 17-Feb-2015.)
Hypotheses
Ref Expression
kur14lem1.a 𝐴𝑋
kur14lem1.c (𝑋𝐴) ∈ 𝑇
kur14lem1.k (𝐾𝐴) ∈ 𝑇
Assertion
Ref Expression
kur14lem1 (𝑁 = 𝐴 → (𝑁𝑋 ∧ {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇))

Proof of Theorem kur14lem1
StepHypRef Expression
1 kur14lem1.a . . 3 𝐴𝑋
2 sseq1 3589 . . 3 (𝑁 = 𝐴 → (𝑁𝑋𝐴𝑋))
31, 2mpbiri 247 . 2 (𝑁 = 𝐴𝑁𝑋)
4 difeq2 3684 . . . 4 (𝑁 = 𝐴 → (𝑋𝑁) = (𝑋𝐴))
5 fveq2 6103 . . . 4 (𝑁 = 𝐴 → (𝐾𝑁) = (𝐾𝐴))
64, 5preq12d 4220 . . 3 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} = {(𝑋𝐴), (𝐾𝐴)})
7 kur14lem1.c . . . 4 (𝑋𝐴) ∈ 𝑇
8 kur14lem1.k . . . 4 (𝐾𝐴) ∈ 𝑇
9 prssi 4293 . . . 4 (((𝑋𝐴) ∈ 𝑇 ∧ (𝐾𝐴) ∈ 𝑇) → {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇)
107, 8, 9mp2an 704 . . 3 {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇
116, 10syl6eqss 3618 . 2 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇)
123, 11jca 553 1 (𝑁 = 𝐴 → (𝑁𝑋 ∧ {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  {cpr 4127  ‘cfv 5804 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-ral 2901  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:  kur14lem7  30448
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