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Theorem kur14lem1 31787
Description: Lemma for kur14 31797. (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 3844 . . 3 (𝑁 = 𝐴 → (𝑁𝑋𝐴𝑋))
31, 2mpbiri 250 . 2 (𝑁 = 𝐴𝑁𝑋)
4 difeq2 3944 . . . 4 (𝑁 = 𝐴 → (𝑋𝑁) = (𝑋𝐴))
5 fveq2 6446 . . . 4 (𝑁 = 𝐴 → (𝐾𝑁) = (𝐾𝐴))
64, 5preq12d 4507 . . 3 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} = {(𝑋𝐴), (𝐾𝐴)})
7 kur14lem1.c . . . 4 (𝑋𝐴) ∈ 𝑇
8 kur14lem1.k . . . 4 (𝐾𝐴) ∈ 𝑇
9 prssi 4583 . . . 4 (((𝑋𝐴) ∈ 𝑇 ∧ (𝐾𝐴) ∈ 𝑇) → {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇)
107, 8, 9mp2an 682 . . 3 {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇
116, 10syl6eqss 3873 . 2 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇)
123, 11jca 507 1 (𝑁 = 𝐴 → (𝑁𝑋 ∧ {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106  cdif 3788  wss 3791  {cpr 4399  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143
This theorem is referenced by:  kur14lem7  31793
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