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Theorem disjpr2 4641
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 4562 . . . 4 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
21ineq2i 4184 . . 3 ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ({𝐴, 𝐵} ∩ ({𝐶} ∪ {𝐷}))
3 indi 4248 . . 3 ({𝐴, 𝐵} ∩ ({𝐶} ∪ {𝐷})) = (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷}))
42, 3eqtri 2842 . 2 ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷}))
5 df-pr 4562 . . . . . . . 8 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65ineq1i 4183 . . . . . . 7 ({𝐴, 𝐵} ∩ {𝐶}) = (({𝐴} ∪ {𝐵}) ∩ {𝐶})
7 indir 4250 . . . . . . 7 (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶}))
86, 7eqtri 2842 . . . . . 6 ({𝐴, 𝐵} ∩ {𝐶}) = (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶}))
9 disjsn2 4640 . . . . . . . 8 (𝐴𝐶 → ({𝐴} ∩ {𝐶}) = ∅)
10 disjsn2 4640 . . . . . . . 8 (𝐵𝐶 → ({𝐵} ∩ {𝐶}) = ∅)
119, 10anim12i 614 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → (({𝐴} ∩ {𝐶}) = ∅ ∧ ({𝐵} ∩ {𝐶}) = ∅))
12 un00 4392 . . . . . . 7 ((({𝐴} ∩ {𝐶}) = ∅ ∧ ({𝐵} ∩ {𝐶}) = ∅) ↔ (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶})) = ∅)
1311, 12sylib 220 . . . . . 6 ((𝐴𝐶𝐵𝐶) → (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶})) = ∅)
148, 13syl5eq 2866 . . . . 5 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
1514adantr 483 . . . 4 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
165ineq1i 4183 . . . . . . 7 ({𝐴, 𝐵} ∩ {𝐷}) = (({𝐴} ∪ {𝐵}) ∩ {𝐷})
17 indir 4250 . . . . . . 7 (({𝐴} ∪ {𝐵}) ∩ {𝐷}) = (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷}))
1816, 17eqtri 2842 . . . . . 6 ({𝐴, 𝐵} ∩ {𝐷}) = (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷}))
19 disjsn2 4640 . . . . . . . 8 (𝐴𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
20 disjsn2 4640 . . . . . . . 8 (𝐵𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
2119, 20anim12i 614 . . . . . . 7 ((𝐴𝐷𝐵𝐷) → (({𝐴} ∩ {𝐷}) = ∅ ∧ ({𝐵} ∩ {𝐷}) = ∅))
22 un00 4392 . . . . . . 7 ((({𝐴} ∩ {𝐷}) = ∅ ∧ ({𝐵} ∩ {𝐷}) = ∅) ↔ (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷})) = ∅)
2321, 22sylib 220 . . . . . 6 ((𝐴𝐷𝐵𝐷) → (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷})) = ∅)
2418, 23syl5eq 2866 . . . . 5 ((𝐴𝐷𝐵𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
2524adantl 484 . . . 4 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
2615, 25uneq12d 4138 . . 3 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷})) = (∅ ∪ ∅))
27 un0 4342 . . 3 (∅ ∪ ∅) = ∅
2826, 27syl6eq 2870 . 2 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷})) = ∅)
294, 28syl5eq 2866 1 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wne 3014  cun 3932  cin 3933  c0 4289  {csn 4559  {cpr 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-pr 4562
This theorem is referenced by:  disjprsn  4642  disjtp2  4644  funcnvqp  6411
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