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Theorem 1wlkdlem2 28022
 Description: Lemma 2 for 1wlkd 28025. (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 4556 . . . . 5 (𝑋𝑉𝑋 ∈ {𝑋})
31, 2syl 17 . . . 4 (𝜑𝑋 ∈ {𝑋})
43adantr 484 . . 3 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ {𝑋})
5 1wlkd.l . . 3 ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})
64, 5eleqtrrd 2855 . 2 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝐼𝐽))
7 1wlkd.j . . . 4 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))
8 1wlkd.y . . . . . 6 (𝜑𝑌𝑉)
98adantr 484 . . . . 5 ((𝜑𝑋𝑌) → 𝑌𝑉)
10 prssg 4709 . . . . 5 ((𝑋𝑉𝑌𝑉) → ((𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼𝐽)))
111, 9, 10syl2an2r 684 . . . 4 ((𝜑𝑋𝑌) → ((𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼𝐽)))
127, 11mpbird 260 . . 3 ((𝜑𝑋𝑌) → (𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)))
1312simpld 498 . 2 ((𝜑𝑋𝑌) → 𝑋 ∈ (𝐼𝐽))
146, 13pm2.61dane 3038 1 (𝜑𝑋 ∈ (𝐼𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   ⊆ wss 3858  {csn 4522  {cpr 4524  ‘cfv 6335  ⟨“cs1 13996  ⟨“cs2 14250 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525 This theorem is referenced by:  1wlkdlem3  28023  1wlkdlem4  28024
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