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Theorem disjpr2OLD 4247
Description: Obsolete proof of disjpr2 4246 as of 23-Jul-2021. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
disjpr2OLD (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)

Proof of Theorem disjpr2OLD
StepHypRef Expression
1 df-pr 4178 . . . 4 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
21a1i 11 . . 3 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}))
32ineq2d 3812 . 2 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ({𝐴, 𝐵} ∩ ({𝐶} ∪ {𝐷})))
4 indi 3871 . . 3 ({𝐴, 𝐵} ∩ ({𝐶} ∪ {𝐷})) = (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷}))
5 df-pr 4178 . . . . . . . 8 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65ineq1i 3808 . . . . . . 7 ({𝐴, 𝐵} ∩ {𝐶}) = (({𝐴} ∪ {𝐵}) ∩ {𝐶})
7 indir 3873 . . . . . . 7 (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶}))
86, 7eqtri 2643 . . . . . 6 ({𝐴, 𝐵} ∩ {𝐶}) = (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶}))
9 disjsn2 4245 . . . . . . . . . 10 (𝐴𝐶 → ({𝐴} ∩ {𝐶}) = ∅)
109adantr 481 . . . . . . . . 9 ((𝐴𝐶𝐵𝐶) → ({𝐴} ∩ {𝐶}) = ∅)
1110adantr 481 . . . . . . . 8 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴} ∩ {𝐶}) = ∅)
12 disjsn2 4245 . . . . . . . . . 10 (𝐵𝐶 → ({𝐵} ∩ {𝐶}) = ∅)
1312adantl 482 . . . . . . . . 9 ((𝐴𝐶𝐵𝐶) → ({𝐵} ∩ {𝐶}) = ∅)
1413adantr 481 . . . . . . . 8 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐵} ∩ {𝐶}) = ∅)
1511, 14jca 554 . . . . . . 7 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴} ∩ {𝐶}) = ∅ ∧ ({𝐵} ∩ {𝐶}) = ∅))
16 un00 4009 . . . . . . 7 ((({𝐴} ∩ {𝐶}) = ∅ ∧ ({𝐵} ∩ {𝐶}) = ∅) ↔ (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶})) = ∅)
1715, 16sylib 208 . . . . . 6 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴} ∩ {𝐶}) ∪ ({𝐵} ∩ {𝐶})) = ∅)
188, 17syl5eq 2667 . . . . 5 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
195ineq1i 3808 . . . . . . 7 ({𝐴, 𝐵} ∩ {𝐷}) = (({𝐴} ∪ {𝐵}) ∩ {𝐷})
20 indir 3873 . . . . . . 7 (({𝐴} ∪ {𝐵}) ∩ {𝐷}) = (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷}))
2119, 20eqtri 2643 . . . . . 6 ({𝐴, 𝐵} ∩ {𝐷}) = (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷}))
22 disjsn2 4245 . . . . . . . . . 10 (𝐴𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
2322adantr 481 . . . . . . . . 9 ((𝐴𝐷𝐵𝐷) → ({𝐴} ∩ {𝐷}) = ∅)
2423adantl 482 . . . . . . . 8 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴} ∩ {𝐷}) = ∅)
25 disjsn2 4245 . . . . . . . . . 10 (𝐵𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
2625adantl 482 . . . . . . . . 9 ((𝐴𝐷𝐵𝐷) → ({𝐵} ∩ {𝐷}) = ∅)
2726adantl 482 . . . . . . . 8 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐵} ∩ {𝐷}) = ∅)
2824, 27jca 554 . . . . . . 7 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴} ∩ {𝐷}) = ∅ ∧ ({𝐵} ∩ {𝐷}) = ∅))
29 un00 4009 . . . . . . 7 ((({𝐴} ∩ {𝐷}) = ∅ ∧ ({𝐵} ∩ {𝐷}) = ∅) ↔ (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷})) = ∅)
3028, 29sylib 208 . . . . . 6 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴} ∩ {𝐷}) ∪ ({𝐵} ∩ {𝐷})) = ∅)
3121, 30syl5eq 2667 . . . . 5 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅)
3218, 31uneq12d 3766 . . . 4 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷})) = (∅ ∪ ∅))
33 un0 3965 . . . 4 (∅ ∪ ∅) = ∅
3432, 33syl6eq 2671 . . 3 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → (({𝐴, 𝐵} ∩ {𝐶}) ∪ ({𝐴, 𝐵} ∩ {𝐷})) = ∅)
354, 34syl5eq 2667 . 2 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ ({𝐶} ∪ {𝐷})) = ∅)
363, 35eqtrd 2655 1 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wne 2793  cun 3570  cin 3571  c0 3913  {csn 4175  {cpr 4177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-sn 4176  df-pr 4178
This theorem is referenced by: (None)
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