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Theorem kur14lem1 32460
Description: Lemma for kur14 32470. (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 3968 . . 3 (𝑁 = 𝐴 → (𝑁𝑋𝐴𝑋))
31, 2mpbiri 261 . 2 (𝑁 = 𝐴𝑁𝑋)
4 difeq2 4069 . . . 4 (𝑁 = 𝐴 → (𝑋𝑁) = (𝑋𝐴))
5 fveq2 6643 . . . 4 (𝑁 = 𝐴 → (𝐾𝑁) = (𝐾𝐴))
64, 5preq12d 4650 . . 3 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} = {(𝑋𝐴), (𝐾𝐴)})
7 kur14lem1.c . . . 4 (𝑋𝐴) ∈ 𝑇
8 kur14lem1.k . . . 4 (𝐾𝐴) ∈ 𝑇
9 prssi 4727 . . . 4 (((𝑋𝐴) ∈ 𝑇 ∧ (𝐾𝐴) ∈ 𝑇) → {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇)
107, 8, 9mp2an 691 . . 3 {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇
116, 10eqsstrdi 3997 . 2 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇)
123, 11jca 515 1 (𝑁 = 𝐴 → (𝑁𝑋 ∧ {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cdif 3907  wss 3910  {cpr 4542  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336
This theorem is referenced by:  kur14lem7  32466
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