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Theorem diftpsn3 3505
 Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
diftpsn3 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})

Proof of Theorem diftpsn3
StepHypRef Expression
1 df-tp 3383 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
21a1i 9 . . 3 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}))
32difeq1d 3061 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}))
4 difundir 3190 . . 3 (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}) = (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶}))
54a1i 9 . 2 ((𝐴𝐶𝐵𝐶) → (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}) = (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})))
6 df-pr 3382 . . . . . . . . 9 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
76a1i 9 . . . . . . . 8 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}))
87ineq1d 3137 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = (({𝐴} ∪ {𝐵}) ∩ {𝐶}))
9 incom 3129 . . . . . . . . 9 (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = ({𝐶} ∩ ({𝐴} ∪ {𝐵}))
10 indi 3184 . . . . . . . . 9 ({𝐶} ∩ ({𝐴} ∪ {𝐵})) = (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵}))
119, 10eqtri 2060 . . . . . . . 8 (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵}))
1211a1i 9 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵})))
13 necom 2289 . . . . . . . . . . 11 (𝐴𝐶𝐶𝐴)
14 disjsn2 3433 . . . . . . . . . . 11 (𝐶𝐴 → ({𝐶} ∩ {𝐴}) = ∅)
1513, 14sylbi 114 . . . . . . . . . 10 (𝐴𝐶 → ({𝐶} ∩ {𝐴}) = ∅)
1615adantr 261 . . . . . . . . 9 ((𝐴𝐶𝐵𝐶) → ({𝐶} ∩ {𝐴}) = ∅)
17 necom 2289 . . . . . . . . . . 11 (𝐵𝐶𝐶𝐵)
18 disjsn2 3433 . . . . . . . . . . 11 (𝐶𝐵 → ({𝐶} ∩ {𝐵}) = ∅)
1917, 18sylbi 114 . . . . . . . . . 10 (𝐵𝐶 → ({𝐶} ∩ {𝐵}) = ∅)
2019adantl 262 . . . . . . . . 9 ((𝐴𝐶𝐵𝐶) → ({𝐶} ∩ {𝐵}) = ∅)
2116, 20uneq12d 3098 . . . . . . . 8 ((𝐴𝐶𝐵𝐶) → (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵})) = (∅ ∪ ∅))
22 unidm 3086 . . . . . . . 8 (∅ ∪ ∅) = ∅
2321, 22syl6eq 2088 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵})) = ∅)
248, 12, 233eqtrd 2076 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
25 disj3 3272 . . . . . 6 (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶}))
2624, 25sylib 127 . . . . 5 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶}))
2726eqcomd 2045 . . . 4 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∖ {𝐶}) = {𝐴, 𝐵})
28 difid 3292 . . . . 5 ({𝐶} ∖ {𝐶}) = ∅
2928a1i 9 . . . 4 ((𝐴𝐶𝐵𝐶) → ({𝐶} ∖ {𝐶}) = ∅)
3027, 29uneq12d 3098 . . 3 ((𝐴𝐶𝐵𝐶) → (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = ({𝐴, 𝐵} ∪ ∅))
31 un0 3251 . . 3 ({𝐴, 𝐵} ∪ ∅) = {𝐴, 𝐵}
3230, 31syl6eq 2088 . 2 ((𝐴𝐶𝐵𝐶) → (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = {𝐴, 𝐵})
333, 5, 323eqtrd 2076 1 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ≠ wne 2204   ∖ cdif 2914   ∪ cun 2915   ∩ cin 2916  ∅c0 3224  {csn 3375  {cpr 3376  {ctp 3377 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-tp 3383 This theorem is referenced by: (None)
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