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Theorem kur14lem1 33165
Description: Lemma for kur14 33175. (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 3947 . . 3 (𝑁 = 𝐴 → (𝑁𝑋𝐴𝑋))
31, 2mpbiri 257 . 2 (𝑁 = 𝐴𝑁𝑋)
4 difeq2 4052 . . . 4 (𝑁 = 𝐴 → (𝑋𝑁) = (𝑋𝐴))
5 fveq2 6776 . . . 4 (𝑁 = 𝐴 → (𝐾𝑁) = (𝐾𝐴))
64, 5preq12d 4679 . . 3 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} = {(𝑋𝐴), (𝐾𝐴)})
7 kur14lem1.c . . . 4 (𝑋𝐴) ∈ 𝑇
8 kur14lem1.k . . . 4 (𝐾𝐴) ∈ 𝑇
9 prssi 4756 . . . 4 (((𝑋𝐴) ∈ 𝑇 ∧ (𝐾𝐴) ∈ 𝑇) → {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇)
107, 8, 9mp2an 689 . . 3 {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇
116, 10eqsstrdi 3976 . 2 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇)
123, 11jca 512 1 (𝑁 = 𝐴 → (𝑁𝑋 ∧ {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  cdif 3885  wss 3888  {cpr 4565  cfv 6435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-iota 6393  df-fv 6443
This theorem is referenced by:  kur14lem7  33171
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