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Theorem 2wlkdlem7 27074
Description: Lemma 7 for 2wlkd 27078. (Contributed by AV, 14-Feb-2021.)
Hypotheses
Ref Expression
2wlkd.p 𝑃 = ⟨“𝐴𝐵𝐶”⟩
2wlkd.f 𝐹 = ⟨“𝐽𝐾”⟩
2wlkd.s (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))
2wlkd.n (𝜑 → (𝐴𝐵𝐵𝐶))
2wlkd.e (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))
Assertion
Ref Expression
2wlkdlem7 (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V))

Proof of Theorem 2wlkdlem7
StepHypRef Expression
1 2wlkd.p . . 3 𝑃 = ⟨“𝐴𝐵𝐶”⟩
2 2wlkd.f . . 3 𝐹 = ⟨“𝐽𝐾”⟩
3 2wlkd.s . . 3 (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))
4 2wlkd.n . . 3 (𝜑 → (𝐴𝐵𝐵𝐶))
5 2wlkd.e . . 3 (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))
61, 2, 3, 4, 52wlkdlem6 27073 . 2 (𝜑 → (𝐵 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾)))
7 elfvex 6384 . . 3 (𝐵 ∈ (𝐼𝐽) → 𝐽 ∈ V)
8 elfvex 6384 . . 3 (𝐵 ∈ (𝐼𝐾) → 𝐾 ∈ V)
97, 8anim12i 591 . 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 1072   = wceq 1632  wcel 2140  wne 2933  Vcvv 3341  wss 3716  {cpr 4324  cfv 6050  ⟨“cs2 13807  ⟨“cs3 13808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-nul 4942  ax-pow 4993
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-dm 5277  df-iota 6013  df-fv 6058
This theorem is referenced by:  2wlkdlem8  27075  2trld  27080
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