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Theorem 2wlkdlem6 27696
Description: Lemma 6 for 2wlkd 27701. (Contributed by AV, 23-Jan-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
2wlkdlem6 (𝜑 → (𝐵 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾)))

Proof of Theorem 2wlkdlem6
StepHypRef Expression
1 2wlkd.e . 2 (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))
2 prcom 4641 . . . . . . . . 9 {𝐴, 𝐵} = {𝐵, 𝐴}
32sseq1i 3971 . . . . . . . 8 ({𝐴, 𝐵} ⊆ (𝐼𝐽) ↔ {𝐵, 𝐴} ⊆ (𝐼𝐽))
43biimpi 219 . . . . . . 7 ({𝐴, 𝐵} ⊆ (𝐼𝐽) → {𝐵, 𝐴} ⊆ (𝐼𝐽))
54adantl 485 . . . . . 6 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → {𝐵, 𝐴} ⊆ (𝐼𝐽))
6 2wlkd.s . . . . . . . 8 (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))
76simp2d 1140 . . . . . . 7 (𝜑𝐵𝑉)
86simp1d 1139 . . . . . . . 8 (𝜑𝐴𝑉)
98adantr 484 . . . . . . 7 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → 𝐴𝑉)
10 prssg 4725 . . . . . . 7 ((𝐵𝑉𝐴𝑉) → ((𝐵 ∈ (𝐼𝐽) ∧ 𝐴 ∈ (𝐼𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼𝐽)))
117, 9, 10syl2an2r 684 . . . . . 6 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → ((𝐵 ∈ (𝐼𝐽) ∧ 𝐴 ∈ (𝐼𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼𝐽)))
125, 11mpbird 260 . . . . 5 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → (𝐵 ∈ (𝐼𝐽) ∧ 𝐴 ∈ (𝐼𝐽)))
1312simpld 498 . . . 4 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → 𝐵 ∈ (𝐼𝐽))
1413ex 416 . . 3 (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) → 𝐵 ∈ (𝐼𝐽)))
15 simpr 488 . . . . . 6 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → {𝐵, 𝐶} ⊆ (𝐼𝐾))
166simp3d 1141 . . . . . . . 8 (𝜑𝐶𝑉)
1716adantr 484 . . . . . . 7 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → 𝐶𝑉)
18 prssg 4725 . . . . . . 7 ((𝐵𝑉𝐶𝑉) → ((𝐵 ∈ (𝐼𝐾) ∧ 𝐶 ∈ (𝐼𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼𝐾)))
197, 17, 18syl2an2r 684 . . . . . 6 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → ((𝐵 ∈ (𝐼𝐾) ∧ 𝐶 ∈ (𝐼𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼𝐾)))
2015, 19mpbird 260 . . . . 5 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → (𝐵 ∈ (𝐼𝐾) ∧ 𝐶 ∈ (𝐼𝐾)))
2120simpld 498 . . . 4 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → 𝐵 ∈ (𝐼𝐾))
2221ex 416 . . 3 (𝜑 → ({𝐵, 𝐶} ⊆ (𝐼𝐾) → 𝐵 ∈ (𝐼𝐾)))
2314, 22anim12d 611 . 2 (𝜑 → (({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → (𝐵 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾))))
241, 23mpd 15 1 (𝜑 → (𝐵 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3007  wss 3910  {cpr 4542  cfv 6328  ⟨“cs2 14182  ⟨“cs3 14183
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-un 3915  df-in 3917  df-ss 3927  df-sn 4541  df-pr 4543
This theorem is referenced by:  2wlkdlem7  27697
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