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Theorem disjtp2 4407
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 4339 . . 3 {𝐷, 𝐸, 𝐹} = ({𝐷, 𝐸} ∪ {𝐹})
21ineq2i 3973 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹}))
3 df-tp 4339 . . . . . 6 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
43ineq1i 3972 . . . . 5 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸})
5 3simpa 1178 . . . . . . . . 9 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴𝐷𝐵𝐷))
6 3simpa 1178 . . . . . . . . 9 ((𝐴𝐸𝐵𝐸𝐶𝐸) → (𝐴𝐸𝐵𝐸))
7 disjpr2 4404 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷) ∧ (𝐴𝐸𝐵𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
85, 6, 7syl2an 589 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
983adant3 1162 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
10 incom 3967 . . . . . . . 8 ({𝐶} ∩ {𝐷, 𝐸}) = ({𝐷, 𝐸} ∩ {𝐶})
11 necom 2990 . . . . . . . . . . . 12 (𝐶𝐷𝐷𝐶)
1211biimpi 207 . . . . . . . . . . 11 (𝐶𝐷𝐷𝐶)
13123ad2ant3 1165 . . . . . . . . . 10 ((𝐴𝐷𝐵𝐷𝐶𝐷) → 𝐷𝐶)
14 necom 2990 . . . . . . . . . . . 12 (𝐶𝐸𝐸𝐶)
1514biimpi 207 . . . . . . . . . . 11 (𝐶𝐸𝐸𝐶)
16153ad2ant3 1165 . . . . . . . . . 10 ((𝐴𝐸𝐵𝐸𝐶𝐸) → 𝐸𝐶)
17 disjprsn 4405 . . . . . . . . . 10 ((𝐷𝐶𝐸𝐶) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
1813, 16, 17syl2an 589 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
19183adant3 1162 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
2010, 19syl5eq 2811 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐶} ∩ {𝐷, 𝐸}) = ∅)
219, 20jca 507 . . . . . 6 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐶} ∩ {𝐷, 𝐸}) = ∅))
22 undisj1 4190 . . . . . 6 ((({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐶} ∩ {𝐷, 𝐸}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸}) = ∅)
2321, 22sylib 209 . . . . 5 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸}) = ∅)
244, 23syl5eq 2811 . . . 4 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅)
25 disjtpsn 4406 . . . . 5 ((𝐴𝐹𝐵𝐹𝐶𝐹) → ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅)
26253ad2ant3 1165 . . . 4 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅)
2724, 26jca 507 . . 3 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅))
28 undisj2 4191 . . 3 ((({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅) ↔ ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹})) = ∅)
2927, 28sylib 209 . 2 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹})) = ∅)
302, 29syl5eq 2811 1 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wne 2937  cun 3730  cin 3731  c0 4079  {csn 4334  {cpr 4336  {ctp 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-sn 4335  df-pr 4337  df-tp 4339
This theorem is referenced by:  cnfldfun  20031
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