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Theorem disjtp2 4284
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 4215 . . 3 {𝐷, 𝐸, 𝐹} = ({𝐷, 𝐸} ∪ {𝐹})
21ineq2i 3844 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹}))
3 df-tp 4215 . . . . . 6 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
43ineq1i 3843 . . . . 5 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸})
5 3simpa 1078 . . . . . . . . 9 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴𝐷𝐵𝐷))
6 3simpa 1078 . . . . . . . . 9 ((𝐴𝐸𝐵𝐸𝐶𝐸) → (𝐴𝐸𝐵𝐸))
7 disjpr2 4280 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷) ∧ (𝐴𝐸𝐵𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
85, 6, 7syl2an 493 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
983adant3 1101 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
10 incom 3838 . . . . . . . 8 ({𝐶} ∩ {𝐷, 𝐸}) = ({𝐷, 𝐸} ∩ {𝐶})
11 necom 2876 . . . . . . . . . . . 12 (𝐶𝐷𝐷𝐶)
1211biimpi 206 . . . . . . . . . . 11 (𝐶𝐷𝐷𝐶)
13123ad2ant3 1104 . . . . . . . . . 10 ((𝐴𝐷𝐵𝐷𝐶𝐷) → 𝐷𝐶)
14 necom 2876 . . . . . . . . . . . 12 (𝐶𝐸𝐸𝐶)
1514biimpi 206 . . . . . . . . . . 11 (𝐶𝐸𝐸𝐶)
16153ad2ant3 1104 . . . . . . . . . 10 ((𝐴𝐸𝐵𝐸𝐶𝐸) → 𝐸𝐶)
17 disjprsn 4282 . . . . . . . . . 10 ((𝐷𝐶𝐸𝐶) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
1813, 16, 17syl2an 493 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
19183adant3 1101 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
2010, 19syl5eq 2697 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐶} ∩ {𝐷, 𝐸}) = ∅)
219, 20jca 553 . . . . . 6 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐶} ∩ {𝐷, 𝐸}) = ∅))
22 undisj1 4062 . . . . . 6 ((({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐶} ∩ {𝐷, 𝐸}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸}) = ∅)
2321, 22sylib 208 . . . . 5 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸}) = ∅)
244, 23syl5eq 2697 . . . 4 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅)
25 disjtpsn 4283 . . . . 5 ((𝐴𝐹𝐵𝐹𝐶𝐹) → ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅)
26253ad2ant3 1104 . . . 4 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅)
2724, 26jca 553 . . 3 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅))
28 undisj2 4063 . . 3 ((({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅) ↔ ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹})) = ∅)
2927, 28sylib 208 . 2 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹})) = ∅)
302, 29syl5eq 2697 1 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wne 2823  cun 3605  cin 3606  c0 3948  {csn 4210  {cpr 4212  {ctp 4214
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-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-tp 4215
This theorem is referenced by:  cnfldfun  19806
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