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Theorem 1wlkdlem2 30157
Description: Lemma 2 for 1wlkd 30160. (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 4660 . . . . 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 2844 . 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 4819 . . . . 5 ((𝑋𝑉𝑌𝑉) → ((𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼𝐽)))
111, 9, 10syl2an2r 685 . . . 4 ((𝜑𝑋𝑌) → ((𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼𝐽)))
127, 11mpbird 257 . . 3 ((𝜑𝑋𝑌) → (𝑋 ∈ (𝐼𝐽) ∧ 𝑌 ∈ (𝐼𝐽)))
1312simpld 494 . 2 ((𝜑𝑋𝑌) → 𝑋 ∈ (𝐼𝐽))
146, 13pm2.61dane 3029 1 (𝜑𝑋 ∈ (𝐼𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wss 3951  {csn 4626  {cpr 4628  cfv 6561  ⟨“cs1 14633  ⟨“cs2 14880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629
This theorem is referenced by:  1wlkdlem3  30158  1wlkdlem4  30159
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