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Theorem disjtp2 4396
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 4326 . . 3 {𝐷, 𝐸, 𝐹} = ({𝐷, 𝐸} ∪ {𝐹})
21ineq2i 3954 . 2 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ({𝐴, 𝐵, 𝐶} ∩ ({𝐷, 𝐸} ∪ {𝐹}))
3 df-tp 4326 . . . . . 6 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
43ineq1i 3953 . . . . 5 ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷, 𝐸})
5 3simpa 1143 . . . . . . . . 9 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴𝐷𝐵𝐷))
6 3simpa 1143 . . . . . . . . 9 ((𝐴𝐸𝐵𝐸𝐶𝐸) → (𝐴𝐸𝐵𝐸))
7 disjpr2 4392 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷) ∧ (𝐴𝐸𝐵𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
85, 6, 7syl2an 495 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
983adant3 1127 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅)
10 incom 3948 . . . . . . . 8 ({𝐶} ∩ {𝐷, 𝐸}) = ({𝐷, 𝐸} ∩ {𝐶})
11 necom 2985 . . . . . . . . . . . 12 (𝐶𝐷𝐷𝐶)
1211biimpi 206 . . . . . . . . . . 11 (𝐶𝐷𝐷𝐶)
13123ad2ant3 1130 . . . . . . . . . 10 ((𝐴𝐷𝐵𝐷𝐶𝐷) → 𝐷𝐶)
14 necom 2985 . . . . . . . . . . . 12 (𝐶𝐸𝐸𝐶)
1514biimpi 206 . . . . . . . . . . 11 (𝐶𝐸𝐸𝐶)
16153ad2ant3 1130 . . . . . . . . . 10 ((𝐴𝐸𝐵𝐸𝐶𝐸) → 𝐸𝐶)
17 disjprsn 4394 . . . . . . . . . 10 ((𝐷𝐶𝐸𝐶) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
1813, 16, 17syl2an 495 . . . . . . . . 9 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
19183adant3 1127 . . . . . . . 8 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐷, 𝐸} ∩ {𝐶}) = ∅)
2010, 19syl5eq 2806 . . . . . . 7 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐶} ∩ {𝐷, 𝐸}) = ∅)
219, 20jca 555 . . . . . 6 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐶} ∩ {𝐷, 𝐸}) = ∅))
22 undisj1 4173 . . . . . 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 2806 . . . 4 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅)
25 disjtpsn 4395 . . . . 5 ((𝐴𝐹𝐵𝐹𝐶𝐹) → ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅)
26253ad2ant3 1130 . . . 4 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅)
2724, 26jca 555 . . 3 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → (({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸}) = ∅ ∧ ({𝐴, 𝐵, 𝐶} ∩ {𝐹}) = ∅))
28 undisj2 4174 . . 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 2806 1 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wne 2932  cun 3713  cin 3714  c0 4058  {csn 4321  {cpr 4323  {ctp 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324  df-tp 4326
This theorem is referenced by:  cnfldfun  19960
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