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Theorem 21wlkdlem7 41139
 Description: Lemma 7 for 21wlkd 41143. (Contributed by AV, 14-Feb-2021.)
Hypotheses
Ref Expression
21wlkd.p 𝑃 = ⟨“𝐴𝐵𝐶”⟩
21wlkd.f 𝐹 = ⟨“𝐽𝐾”⟩
21wlkd.s (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))
21wlkd.n (𝜑 → (𝐴𝐵𝐵𝐶))
21wlkd.e (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))
Assertion
Ref Expression
21wlkdlem7 (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V))

Proof of Theorem 21wlkdlem7
StepHypRef Expression
1 21wlkd.p . . 3 𝑃 = ⟨“𝐴𝐵𝐶”⟩
2 21wlkd.f . . 3 𝐹 = ⟨“𝐽𝐾”⟩
3 21wlkd.s . . 3 (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))
4 21wlkd.n . . 3 (𝜑 → (𝐴𝐵𝐵𝐶))
5 21wlkd.e . . 3 (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))
61, 2, 3, 4, 521wlkdlem6 41138 . 2 (𝜑 → (𝐵 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾)))
7 elfvex 6131 . . 3 (𝐵 ∈ (𝐼𝐽) → 𝐽 ∈ V)
8 elfvex 6131 . . 3 (𝐵 ∈ (𝐼𝐾) → 𝐾 ∈ V)
97, 8anim12i 588 . 2 ((𝐵 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾)) → (𝐽 ∈ V ∧ 𝐾 ∈ V))
106, 9syl 17 1 (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ⊆ wss 3540  {cpr 4127  ‘cfv 5804  ⟨“cs2 13437  ⟨“cs3 13438 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812 This theorem is referenced by:  21wlkdlem8  41140  2trld  41145
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