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Theorem disjpr2 3425
Description: The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
Assertion
Ref Expression
disjpr2 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({A, B} ∩ {𝐶, 𝐷}) = ∅)

Proof of Theorem disjpr2
StepHypRef Expression
1 df-pr 3374 . . . 4 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
21a1i 9 . . 3 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}))
32ineq2d 3132 . 2 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({A, B} ∩ {𝐶, 𝐷}) = ({A, B} ∩ ({𝐶} ∪ {𝐷})))
4 indi 3178 . . 3 ({A, B} ∩ ({𝐶} ∪ {𝐷})) = (({A, B} ∩ {𝐶}) ∪ ({A, B} ∩ {𝐷}))
5 df-pr 3374 . . . . . . . 8 {A, B} = ({A} ∪ {B})
65ineq1i 3128 . . . . . . 7 ({A, B} ∩ {𝐶}) = (({A} ∪ {B}) ∩ {𝐶})
7 indir 3180 . . . . . . 7 (({A} ∪ {B}) ∩ {𝐶}) = (({A} ∩ {𝐶}) ∪ ({B} ∩ {𝐶}))
86, 7eqtri 2057 . . . . . 6 ({A, B} ∩ {𝐶}) = (({A} ∩ {𝐶}) ∪ ({B} ∩ {𝐶}))
9 disjsn2 3424 . . . . . . . . . 10 (A𝐶 → ({A} ∩ {𝐶}) = ∅)
109adantr 261 . . . . . . . . 9 ((A𝐶 B𝐶) → ({A} ∩ {𝐶}) = ∅)
1110adantr 261 . . . . . . . 8 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({A} ∩ {𝐶}) = ∅)
12 disjsn2 3424 . . . . . . . . . 10 (B𝐶 → ({B} ∩ {𝐶}) = ∅)
1312adantl 262 . . . . . . . . 9 ((A𝐶 B𝐶) → ({B} ∩ {𝐶}) = ∅)
1413adantr 261 . . . . . . . 8 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({B} ∩ {𝐶}) = ∅)
1511, 14jca 290 . . . . . . 7 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → (({A} ∩ {𝐶}) = ∅ ({B} ∩ {𝐶}) = ∅))
16 un00 3257 . . . . . . 7 ((({A} ∩ {𝐶}) = ∅ ({B} ∩ {𝐶}) = ∅) ↔ (({A} ∩ {𝐶}) ∪ ({B} ∩ {𝐶})) = ∅)
1715, 16sylib 127 . . . . . 6 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → (({A} ∩ {𝐶}) ∪ ({B} ∩ {𝐶})) = ∅)
188, 17syl5eq 2081 . . . . 5 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({A, B} ∩ {𝐶}) = ∅)
195ineq1i 3128 . . . . . . 7 ({A, B} ∩ {𝐷}) = (({A} ∪ {B}) ∩ {𝐷})
20 indir 3180 . . . . . . 7 (({A} ∪ {B}) ∩ {𝐷}) = (({A} ∩ {𝐷}) ∪ ({B} ∩ {𝐷}))
2119, 20eqtri 2057 . . . . . 6 ({A, B} ∩ {𝐷}) = (({A} ∩ {𝐷}) ∪ ({B} ∩ {𝐷}))
22 disjsn2 3424 . . . . . . . . . 10 (A𝐷 → ({A} ∩ {𝐷}) = ∅)
2322adantr 261 . . . . . . . . 9 ((A𝐷 B𝐷) → ({A} ∩ {𝐷}) = ∅)
2423adantl 262 . . . . . . . 8 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({A} ∩ {𝐷}) = ∅)
25 disjsn2 3424 . . . . . . . . . 10 (B𝐷 → ({B} ∩ {𝐷}) = ∅)
2625adantl 262 . . . . . . . . 9 ((A𝐷 B𝐷) → ({B} ∩ {𝐷}) = ∅)
2726adantl 262 . . . . . . . 8 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({B} ∩ {𝐷}) = ∅)
2824, 27jca 290 . . . . . . 7 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → (({A} ∩ {𝐷}) = ∅ ({B} ∩ {𝐷}) = ∅))
29 un00 3257 . . . . . . 7 ((({A} ∩ {𝐷}) = ∅ ({B} ∩ {𝐷}) = ∅) ↔ (({A} ∩ {𝐷}) ∪ ({B} ∩ {𝐷})) = ∅)
3028, 29sylib 127 . . . . . 6 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → (({A} ∩ {𝐷}) ∪ ({B} ∩ {𝐷})) = ∅)
3121, 30syl5eq 2081 . . . . 5 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({A, B} ∩ {𝐷}) = ∅)
3218, 31uneq12d 3092 . . . 4 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → (({A, B} ∩ {𝐶}) ∪ ({A, B} ∩ {𝐷})) = (∅ ∪ ∅))
33 un0 3245 . . . 4 (∅ ∪ ∅) = ∅
3432, 33syl6eq 2085 . . 3 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → (({A, B} ∩ {𝐶}) ∪ ({A, B} ∩ {𝐷})) = ∅)
354, 34syl5eq 2081 . 2 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({A, B} ∩ ({𝐶} ∪ {𝐷})) = ∅)
363, 35eqtrd 2069 1 (((A𝐶 B𝐶) (A𝐷 B𝐷)) → ({A, B} ∩ {𝐶, 𝐷}) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wne 2201  cun 2909  cin 2910  c0 3218  {csn 3367  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374
This theorem is referenced by: (None)
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