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Theorem 2wlkdlem6 26896
Description: Lemma 6 for 2wlkd 26901. (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 4299 . . . . . . . . 9 {𝐴, 𝐵} = {𝐵, 𝐴}
32sseq1i 3662 . . . . . . . 8 ({𝐴, 𝐵} ⊆ (𝐼𝐽) ↔ {𝐵, 𝐴} ⊆ (𝐼𝐽))
43biimpi 206 . . . . . . 7 ({𝐴, 𝐵} ⊆ (𝐼𝐽) → {𝐵, 𝐴} ⊆ (𝐼𝐽))
54adantl 481 . . . . . 6 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → {𝐵, 𝐴} ⊆ (𝐼𝐽))
6 2wlkd.s . . . . . . . 8 (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))
76simp2d 1094 . . . . . . 7 (𝜑𝐵𝑉)
86simp1d 1093 . . . . . . . 8 (𝜑𝐴𝑉)
98adantr 480 . . . . . . 7 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → 𝐴𝑉)
10 prssg 4382 . . . . . . 7 ((𝐵𝑉𝐴𝑉) → ((𝐵 ∈ (𝐼𝐽) ∧ 𝐴 ∈ (𝐼𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼𝐽)))
117, 9, 10syl2an2r 893 . . . . . 6 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → ((𝐵 ∈ (𝐼𝐽) ∧ 𝐴 ∈ (𝐼𝐽)) ↔ {𝐵, 𝐴} ⊆ (𝐼𝐽)))
125, 11mpbird 247 . . . . 5 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → (𝐵 ∈ (𝐼𝐽) ∧ 𝐴 ∈ (𝐼𝐽)))
1312simpld 474 . . . 4 ((𝜑 ∧ {𝐴, 𝐵} ⊆ (𝐼𝐽)) → 𝐵 ∈ (𝐼𝐽))
1413ex 449 . . 3 (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) → 𝐵 ∈ (𝐼𝐽)))
15 simpr 476 . . . . . 6 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → {𝐵, 𝐶} ⊆ (𝐼𝐾))
166simp3d 1095 . . . . . . . 8 (𝜑𝐶𝑉)
1716adantr 480 . . . . . . 7 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → 𝐶𝑉)
18 prssg 4382 . . . . . . 7 ((𝐵𝑉𝐶𝑉) → ((𝐵 ∈ (𝐼𝐾) ∧ 𝐶 ∈ (𝐼𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼𝐾)))
197, 17, 18syl2an2r 893 . . . . . 6 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → ((𝐵 ∈ (𝐼𝐾) ∧ 𝐶 ∈ (𝐼𝐾)) ↔ {𝐵, 𝐶} ⊆ (𝐼𝐾)))
2015, 19mpbird 247 . . . . 5 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → (𝐵 ∈ (𝐼𝐾) ∧ 𝐶 ∈ (𝐼𝐾)))
2120simpld 474 . . . 4 ((𝜑 ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)) → 𝐵 ∈ (𝐼𝐾))
2221ex 449 . . 3 (𝜑 → ({𝐵, 𝐶} ⊆ (𝐼𝐾) → 𝐵 ∈ (𝐼𝐾)))
2314, 22anim12d 585 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wss 3607  {cpr 4212  cfv 5926  ⟨“cs2 13632  ⟨“cs3 13633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-pr 4213
This theorem is referenced by:  2wlkdlem7  26897
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