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Theorem diftpsn3 3849
 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 3743 . . . 4 {A, B, C} = ({A, B} ∪ {C})
21a1i 10 . . 3 ((AC BC) → {A, B, C} = ({A, B} ∪ {C}))
32difeq1d 3384 . 2 ((AC BC) → ({A, B, C} {C}) = (({A, B} ∪ {C}) {C}))
4 difundir 3508 . . 3 (({A, B} ∪ {C}) {C}) = (({A, B} {C}) ∪ ({C} {C}))
54a1i 10 . 2 ((AC BC) → (({A, B} ∪ {C}) {C}) = (({A, B} {C}) ∪ ({C} {C})))
6 df-pr 3742 . . . . . . . . 9 {A, B} = ({A} ∪ {B})
76a1i 10 . . . . . . . 8 ((AC BC) → {A, B} = ({A} ∪ {B}))
87ineq1d 3456 . . . . . . 7 ((AC BC) → ({A, B} ∩ {C}) = (({A} ∪ {B}) ∩ {C}))
9 incom 3448 . . . . . . . . 9 (({A} ∪ {B}) ∩ {C}) = ({C} ∩ ({A} ∪ {B}))
10 indi 3501 . . . . . . . . 9 ({C} ∩ ({A} ∪ {B})) = (({C} ∩ {A}) ∪ ({C} ∩ {B}))
119, 10eqtri 2373 . . . . . . . 8 (({A} ∪ {B}) ∩ {C}) = (({C} ∩ {A}) ∪ ({C} ∩ {B}))
1211a1i 10 . . . . . . 7 ((AC BC) → (({A} ∪ {B}) ∩ {C}) = (({C} ∩ {A}) ∪ ({C} ∩ {B})))
13 necom 2597 . . . . . . . . . . 11 (ACCA)
14 disjsn2 3787 . . . . . . . . . . 11 (CA → ({C} ∩ {A}) = )
1513, 14sylbi 187 . . . . . . . . . 10 (AC → ({C} ∩ {A}) = )
1615adantr 451 . . . . . . . . 9 ((AC BC) → ({C} ∩ {A}) = )
17 necom 2597 . . . . . . . . . . 11 (BCCB)
18 disjsn2 3787 . . . . . . . . . . 11 (CB → ({C} ∩ {B}) = )
1917, 18sylbi 187 . . . . . . . . . 10 (BC → ({C} ∩ {B}) = )
2019adantl 452 . . . . . . . . 9 ((AC BC) → ({C} ∩ {B}) = )
2116, 20uneq12d 3419 . . . . . . . 8 ((AC BC) → (({C} ∩ {A}) ∪ ({C} ∩ {B})) = ())
22 unidm 3407 . . . . . . . 8 () =
2321, 22syl6eq 2401 . . . . . . 7 ((AC BC) → (({C} ∩ {A}) ∪ ({C} ∩ {B})) = )
248, 12, 233eqtrd 2389 . . . . . 6 ((AC BC) → ({A, B} ∩ {C}) = )
25 disj3 3595 . . . . . 6 (({A, B} ∩ {C}) = ↔ {A, B} = ({A, B} {C}))
2624, 25sylib 188 . . . . 5 ((AC BC) → {A, B} = ({A, B} {C}))
2726eqcomd 2358 . . . 4 ((AC BC) → ({A, B} {C}) = {A, B})
28 difid 3618 . . . . 5 ({C} {C}) =
2928a1i 10 . . . 4 ((AC BC) → ({C} {C}) = )
3027, 29uneq12d 3419 . . 3 ((AC BC) → (({A, B} {C}) ∪ ({C} {C})) = ({A, B} ∪ ))
31 un0 3575 . . 3 ({A, B} ∪ ) = {A, B}
3230, 31syl6eq 2401 . 2 ((AC BC) → (({A, B} {C}) ∪ ({C} {C})) = {A, B})
333, 5, 323eqtrd 2389 1 ((AC BC) → ({A, B, C} {C}) = {A, B})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  {cpr 3738  {ctp 3739 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-tp 3743 This theorem is referenced by: (None)
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