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Theorem disjtp2 4720
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 4633 . . 3 {𝐷, 𝐸, 𝐹} = ({𝐷, 𝐸} ∪ {𝐹})
21ineq2i 4209 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹}))
3 df-tp 4633 . . . . . 6 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
43ineq1i 4208 . . . . 5 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸})
5 3simpa 1148 . . . . . . . . 9 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴𝐷𝐵𝐷))
6 3simpa 1148 . . . . . . . . 9 ((𝐴𝐸𝐵𝐸𝐶𝐸) → (𝐴𝐸𝐵𝐸))
7 disjpr2 4717 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷) ∧ (𝐴𝐸𝐵𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
85, 6, 7syl2an 596 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
983adant3 1132 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
10 incom 4201 . . . . . . . 8 ({𝐶} ∩ {𝐷, 𝐸}) = ({𝐷, 𝐸} ∩ {𝐶})
11 necom 2994 . . . . . . . . . . . 12 (𝐶𝐷𝐷𝐶)
1211biimpi 215 . . . . . . . . . . 11 (𝐶𝐷𝐷𝐶)
13123ad2ant3 1135 . . . . . . . . . 10 ((𝐴𝐷𝐵𝐷𝐶𝐷) → 𝐷𝐶)
14 necom 2994 . . . . . . . . . . . 12 (𝐶𝐸𝐸𝐶)
1514biimpi 215 . . . . . . . . . . 11 (𝐶𝐸𝐸𝐶)
16153ad2ant3 1135 . . . . . . . . . 10 ((𝐴𝐸𝐵𝐸𝐶𝐸) → 𝐸𝐶)
17 disjprsn 4718 . . . . . . . . . 10 ((𝐷𝐶𝐸𝐶) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
1813, 16, 17syl2an 596 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
19183adant3 1132 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
2010, 19eqtrid 2784 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐶} ∩ {𝐷, 𝐸}) = ∅)
219, 20jca 512 . . . . . 6 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐶} ∩ {𝐷, 𝐸}) = ∅))
22 undisj1 4461 . . . . . 6 ((({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐶} ∩ {𝐷, 𝐸}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸}) = ∅)
2321, 22sylib 217 . . . . 5 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸}) = ∅)
244, 23eqtrid 2784 . . . 4 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅)
25 disjtpsn 4719 . . . . 5 ((𝐴𝐹𝐵𝐹𝐶𝐹) → ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅)
26253ad2ant3 1135 . . . 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 4462 . . 3 ((({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅) ↔ ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹})) = ∅)
2927, 28sylib 217 . 2 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹})) = ∅)
302, 29eqtrid 2784 1 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wne 2940  cun 3946  cin 3947  c0 4322  {csn 4628  {cpr 4630  {ctp 4632
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-tp 4633
This theorem is referenced by:  cnfldfunALT  20956  cnfldfunALTOLD  20957
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