Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  kur14lem1 Structured version   Visualization version   GIF version

Theorem kur14lem1 30287
Description: Lemma for kur14 30297. (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 3493 . . 3 (𝑁 = 𝐴 → (𝑁𝑋𝐴𝑋))
31, 2mpbiri 246 . 2 (𝑁 = 𝐴𝑁𝑋)
4 difeq2 3588 . . . 4 (𝑁 = 𝐴 → (𝑋𝑁) = (𝑋𝐴))
5 fveq2 5987 . . . 4 (𝑁 = 𝐴 → (𝐾𝑁) = (𝐾𝐴))
64, 5preq12d 4123 . . 3 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} = {(𝑋𝐴), (𝐾𝐴)})
7 kur14lem1.c . . . 4 (𝑋𝐴) ∈ 𝑇
8 kur14lem1.k . . . 4 (𝐾𝐴) ∈ 𝑇
9 prssi 4196 . . . 4 (((𝑋𝐴) ∈ 𝑇 ∧ (𝐾𝐴) ∈ 𝑇) → {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇)
107, 8, 9mp2an 703 . . 3 {(𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇
116, 10syl6eqss 3522 . 2 (𝑁 = 𝐴 → {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇)
123, 11jca 552 1 (𝑁 = 𝐴 → (𝑁𝑋 ∧ {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  cdif 3441  wss 3444  {cpr 4030  cfv 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
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 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-iota 5653  df-fv 5697
This theorem is referenced by:  kur14lem7  30293
  Copyright terms: Public domain W3C validator