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Theorem 1wlkdlem2 29655
Description: Lemma 2 for 1wlkd 29658. (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 4663 . . . . 5 (𝑋𝑉𝑋 ∈ {𝑋})
31, 2syl 17 . . . 4 (𝜑𝑋 ∈ {𝑋})
43adantr 480 . . 3 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ {𝑋})
5 1wlkd.l . . 3 ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})
64, 5eleqtrrd 2835 . 2 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝐼𝐽))
7 1wlkd.j . . . 4 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))
8 1wlkd.y . . . . . 6 (𝜑𝑌𝑉)
98adantr 480 . . . . 5 ((𝜑𝑋𝑌) → 𝑌𝑉)
10 prssg 4823 . . . . 5 ((𝑋𝑉𝑌𝑉) → ((𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼𝐽)))
111, 9, 10syl2an2r 682 . . . 4 ((𝜑𝑋𝑌) → ((𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼𝐽)))
127, 11mpbird 256 . . 3 ((𝜑𝑋𝑌) → (𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)))
1312simpld 494 . 2 ((𝜑𝑋𝑌) → 𝑋 ∈ (𝐼𝐽))
146, 13pm2.61dane 3028 1 (𝜑𝑋 ∈ (𝐼𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wne 2939  wss 3949  {csn 4629  {cpr 4631  cfv 6544  ⟨“cs1 14550  ⟨“cs2 14797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632
This theorem is referenced by:  1wlkdlem3  29656  1wlkdlem4  29657
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