MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1wlkdlem2 Structured version   Visualization version   GIF version

Theorem 1wlkdlem2 29429
Description: Lemma 2 for 1wlkd 29432. (Contributed by AV, 22-Jan-2021.)
Hypotheses
Ref Expression
1wlkd.p 𝑃 = ⟨“𝑋𝑌”⟩
1wlkd.f 𝐹 = ⟨“𝐽”⟩
1wlkd.x (𝜑𝑋𝑉)
1wlkd.y (𝜑𝑌𝑉)
1wlkd.l ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})
1wlkd.j ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))
Assertion
Ref Expression
1wlkdlem2 (𝜑𝑋 ∈ (𝐼𝐽))

Proof of Theorem 1wlkdlem2
StepHypRef Expression
1 1wlkd.x . . . . 5 (𝜑𝑋𝑉)
2 snidg 4662 . . . . 5 (𝑋𝑉𝑋 ∈ {𝑋})
31, 2syl 17 . . . 4 (𝜑𝑋 ∈ {𝑋})
43adantr 481 . . 3 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ {𝑋})
5 1wlkd.l . . 3 ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})
64, 5eleqtrrd 2836 . 2 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝐼𝐽))
7 1wlkd.j . . . 4 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))
8 1wlkd.y . . . . . 6 (𝜑𝑌𝑉)
98adantr 481 . . . . 5 ((𝜑𝑋𝑌) → 𝑌𝑉)
10 prssg 4822 . . . . 5 ((𝑋𝑉𝑌𝑉) → ((𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼𝐽)))
111, 9, 10syl2an2r 683 . . . 4 ((𝜑𝑋𝑌) → ((𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼𝐽)))
127, 11mpbird 256 . . 3 ((𝜑𝑋𝑌) → (𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)))
1312simpld 495 . 2 ((𝜑𝑋𝑌) → 𝑋 ∈ (𝐼𝐽))
146, 13pm2.61dane 3029 1 (𝜑𝑋 ∈ (𝐼𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  wss 3948  {csn 4628  {cpr 4630  cfv 6543  ⟨“cs1 14547  ⟨“cs2 14794
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631
This theorem is referenced by:  1wlkdlem3  29430  1wlkdlem4  29431
  Copyright terms: Public domain W3C validator