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Theorem disjtp2 4644
Description: Two completely distinct unordered triples are disjoint. (Contributed by AV, 14-Nov-2021.)
Assertion
Ref Expression
disjtp2 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅)

Proof of Theorem disjtp2
StepHypRef Expression
1 df-tp 4562 . . 3 {𝐷, 𝐸, 𝐹} = ({𝐷, 𝐸} ∪ {𝐹})
21ineq2i 4183 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹}))
3 df-tp 4562 . . . . . 6 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
43ineq1i 4182 . . . . 5 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸})
5 3simpa 1140 . . . . . . . . 9 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴𝐷𝐵𝐷))
6 3simpa 1140 . . . . . . . . 9 ((𝐴𝐸𝐵𝐸𝐶𝐸) → (𝐴𝐸𝐵𝐸))
7 disjpr2 4641 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷) ∧ (𝐴𝐸𝐵𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
85, 6, 7syl2an 595 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
983adant3 1124 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
10 incom 4175 . . . . . . . 8 ({𝐶} ∩ {𝐷, 𝐸}) = ({𝐷, 𝐸} ∩ {𝐶})
11 necom 3066 . . . . . . . . . . . 12 (𝐶𝐷𝐷𝐶)
1211biimpi 217 . . . . . . . . . . 11 (𝐶𝐷𝐷𝐶)
13123ad2ant3 1127 . . . . . . . . . 10 ((𝐴𝐷𝐵𝐷𝐶𝐷) → 𝐷𝐶)
14 necom 3066 . . . . . . . . . . . 12 (𝐶𝐸𝐸𝐶)
1514biimpi 217 . . . . . . . . . . 11 (𝐶𝐸𝐸𝐶)
16153ad2ant3 1127 . . . . . . . . . 10 ((𝐴𝐸𝐵𝐸𝐶𝐸) → 𝐸𝐶)
17 disjprsn 4642 . . . . . . . . . 10 ((𝐷𝐶𝐸𝐶) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
1813, 16, 17syl2an 595 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
19183adant3 1124 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
2010, 19syl5eq 2865 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐶} ∩ {𝐷, 𝐸}) = ∅)
219, 20jca 512 . . . . . 6 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐶} ∩ {𝐷, 𝐸}) = ∅))
22 undisj1 4407 . . . . . 6 ((({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐶} ∩ {𝐷, 𝐸}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸}) = ∅)
2321, 22sylib 219 . . . . 5 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸}) = ∅)
244, 23syl5eq 2865 . . . 4 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅)
25 disjtpsn 4643 . . . . 5 ((𝐴𝐹𝐵𝐹𝐶𝐹) → ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅)
26253ad2ant3 1127 . . . 4 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅)
2724, 26jca 512 . . 3 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅))
28 undisj2 4408 . . 3 ((({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅) ↔ ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹})) = ∅)
2927, 28sylib 219 . 2 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹})) = ∅)
302, 29syl5eq 2865 1 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wne 3013  cun 3931  cin 3932  c0 4288  {csn 4557  {cpr 4559  {ctp 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560  df-tp 4562
This theorem is referenced by:  cnfldfun  20485
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