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Theorem diftpsn3 3496
Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
diftpsn3 ((A𝐶 B𝐶) → ({A, B, 𝐶} ∖ {𝐶}) = {A, B})

Proof of Theorem diftpsn3
StepHypRef Expression
1 df-tp 3375 . . . 4 {A, B, 𝐶} = ({A, B} ∪ {𝐶})
21a1i 9 . . 3 ((A𝐶 B𝐶) → {A, B, 𝐶} = ({A, B} ∪ {𝐶}))
32difeq1d 3055 . 2 ((A𝐶 B𝐶) → ({A, B, 𝐶} ∖ {𝐶}) = (({A, B} ∪ {𝐶}) ∖ {𝐶}))
4 difundir 3184 . . 3 (({A, B} ∪ {𝐶}) ∖ {𝐶}) = (({A, B} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶}))
54a1i 9 . 2 ((A𝐶 B𝐶) → (({A, B} ∪ {𝐶}) ∖ {𝐶}) = (({A, B} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})))
6 df-pr 3374 . . . . . . . . 9 {A, B} = ({A} ∪ {B})
76a1i 9 . . . . . . . 8 ((A𝐶 B𝐶) → {A, B} = ({A} ∪ {B}))
87ineq1d 3131 . . . . . . 7 ((A𝐶 B𝐶) → ({A, B} ∩ {𝐶}) = (({A} ∪ {B}) ∩ {𝐶}))
9 incom 3123 . . . . . . . . 9 (({A} ∪ {B}) ∩ {𝐶}) = ({𝐶} ∩ ({A} ∪ {B}))
10 indi 3178 . . . . . . . . 9 ({𝐶} ∩ ({A} ∪ {B})) = (({𝐶} ∩ {A}) ∪ ({𝐶} ∩ {B}))
119, 10eqtri 2057 . . . . . . . 8 (({A} ∪ {B}) ∩ {𝐶}) = (({𝐶} ∩ {A}) ∪ ({𝐶} ∩ {B}))
1211a1i 9 . . . . . . 7 ((A𝐶 B𝐶) → (({A} ∪ {B}) ∩ {𝐶}) = (({𝐶} ∩ {A}) ∪ ({𝐶} ∩ {B})))
13 necom 2283 . . . . . . . . . . 11 (A𝐶𝐶A)
14 disjsn2 3424 . . . . . . . . . . 11 (𝐶A → ({𝐶} ∩ {A}) = ∅)
1513, 14sylbi 114 . . . . . . . . . 10 (A𝐶 → ({𝐶} ∩ {A}) = ∅)
1615adantr 261 . . . . . . . . 9 ((A𝐶 B𝐶) → ({𝐶} ∩ {A}) = ∅)
17 necom 2283 . . . . . . . . . . 11 (B𝐶𝐶B)
18 disjsn2 3424 . . . . . . . . . . 11 (𝐶B → ({𝐶} ∩ {B}) = ∅)
1917, 18sylbi 114 . . . . . . . . . 10 (B𝐶 → ({𝐶} ∩ {B}) = ∅)
2019adantl 262 . . . . . . . . 9 ((A𝐶 B𝐶) → ({𝐶} ∩ {B}) = ∅)
2116, 20uneq12d 3092 . . . . . . . 8 ((A𝐶 B𝐶) → (({𝐶} ∩ {A}) ∪ ({𝐶} ∩ {B})) = (∅ ∪ ∅))
22 unidm 3080 . . . . . . . 8 (∅ ∪ ∅) = ∅
2321, 22syl6eq 2085 . . . . . . 7 ((A𝐶 B𝐶) → (({𝐶} ∩ {A}) ∪ ({𝐶} ∩ {B})) = ∅)
248, 12, 233eqtrd 2073 . . . . . 6 ((A𝐶 B𝐶) → ({A, B} ∩ {𝐶}) = ∅)
25 disj3 3266 . . . . . 6 (({A, B} ∩ {𝐶}) = ∅ ↔ {A, B} = ({A, B} ∖ {𝐶}))
2624, 25sylib 127 . . . . 5 ((A𝐶 B𝐶) → {A, B} = ({A, B} ∖ {𝐶}))
2726eqcomd 2042 . . . 4 ((A𝐶 B𝐶) → ({A, B} ∖ {𝐶}) = {A, B})
28 difid 3286 . . . . 5 ({𝐶} ∖ {𝐶}) = ∅
2928a1i 9 . . . 4 ((A𝐶 B𝐶) → ({𝐶} ∖ {𝐶}) = ∅)
3027, 29uneq12d 3092 . . 3 ((A𝐶 B𝐶) → (({A, B} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = ({A, B} ∪ ∅))
31 un0 3245 . . 3 ({A, B} ∪ ∅) = {A, B}
3230, 31syl6eq 2085 . 2 ((A𝐶 B𝐶) → (({A, B} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = {A, B})
333, 5, 323eqtrd 2073 1 ((A𝐶 B𝐶) → ({A, B, 𝐶} ∖ {𝐶}) = {A, B})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wne 2201  cdif 2908  cun 2909  cin 2910  c0 3218  {csn 3367  {cpr 3368  {ctp 3369
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-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-tp 3375
This theorem is referenced by: (None)
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