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Theorem disjpr2 4718
Description: Two completely distinct unordered pairs are disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Proof shortened by JJ, 23-Jul-2021.)
Assertion
Ref Expression
disjpr2 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)

Proof of Theorem disjpr2
StepHypRef Expression
1 df-pr 4634 . . . 4 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
21ineq2i 4225 . . 3 ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ({𝐴, 𝐵} ∩ ({𝐶} ∪ {𝐷}))
3 indi 4290 . . 3 ({𝐴, 𝐵} ∩ ({𝐶} ∪ {𝐷})) = (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷}))
42, 3eqtri 2763 . 2 ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷}))
5 df-pr 4634 . . . . . . . 8 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65ineq1i 4224 . . . . . . 7 ({𝐴, 𝐵} ∩ {𝐶}) = (({𝐴} ∪ {𝐵}) ∩ {𝐶})
7 indir 4292 . . . . . . 7 (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶}))
86, 7eqtri 2763 . . . . . 6 ({𝐴, 𝐵} ∩ {𝐶}) = (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶}))
9 disjsn2 4717 . . . . . . . 8 (𝐴𝐶 → ({𝐴} ∩ {𝐶}) = ∅)
10 disjsn2 4717 . . . . . . . 8 (𝐵𝐶 → ({𝐵} ∩ {𝐶}) = ∅)
119, 10anim12i 613 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → (({𝐴} ∩ {𝐶}) = ∅ ∧ ({𝐵} ∩ {𝐶}) = ∅))
12 un00 4451 . . . . . . 7 ((({𝐴} ∩ {𝐶}) = ∅ ∧ ({𝐵} ∩ {𝐶}) = ∅) ↔ (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶})) = ∅)
1311, 12sylib 218 . . . . . 6 ((𝐴𝐶𝐵𝐶) → (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶})) = ∅)
148, 13eqtrid 2787 . . . . 5 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
1514adantr 480 . . . 4 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
165ineq1i 4224 . . . . . . 7 ({𝐴, 𝐵} ∩ {𝐷}) = (({𝐴} ∪ {𝐵}) ∩ {𝐷})
17 indir 4292 . . . . . . 7 (({𝐴} ∪ {𝐵}) ∩ {𝐷}) = (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷}))
1816, 17eqtri 2763 . . . . . 6 ({𝐴, 𝐵} ∩ {𝐷}) = (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷}))
19 disjsn2 4717 . . . . . . . 8 (𝐴𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
20 disjsn2 4717 . . . . . . . 8 (𝐵𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
2119, 20anim12i 613 . . . . . . 7 ((𝐴𝐷𝐵𝐷) → (({𝐴} ∩ {𝐷}) = ∅ ∧ ({𝐵} ∩ {𝐷}) = ∅))
22 un00 4451 . . . . . . 7 ((({𝐴} ∩ {𝐷}) = ∅ ∧ ({𝐵} ∩ {𝐷}) = ∅) ↔ (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷})) = ∅)
2321, 22sylib 218 . . . . . 6 ((𝐴𝐷𝐵𝐷) → (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷})) = ∅)
2418, 23eqtrid 2787 . . . . 5 ((𝐴𝐷𝐵𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
2524adantl 481 . . . 4 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
2615, 25uneq12d 4179 . . 3 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷})) = (∅ ∪ ∅))
27 un0 4400 . . 3 (∅ ∪ ∅) = ∅
2826, 27eqtrdi 2791 . 2 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷})) = ∅)
294, 28eqtrid 2787 1 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wne 2938  cun 3961  cin 3962  c0 4339  {csn 4631  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634
This theorem is referenced by:  disjprsn  4719  disjtp2  4721  funcnvqp  6632  evl1deg3  33583
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