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Theorem diftpsn3OLD 4307
Description: Obsolete proof of diftpsn3 4306 as of 23-Jul-2021. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
diftpsn3OLD ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})

Proof of Theorem diftpsn3OLD
StepHypRef Expression
1 df-tp 4158 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
21a1i 11 . . 3 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}))
32difeq1d 3710 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}))
4 difundir 3861 . . 3 (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}) = (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶}))
54a1i 11 . 2 ((𝐴𝐶𝐵𝐶) → (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}) = (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})))
6 df-pr 4156 . . . . . . . . 9 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
76a1i 11 . . . . . . . 8 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}))
87ineq1d 3796 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = (({𝐴} ∪ {𝐵}) ∩ {𝐶}))
9 incom 3788 . . . . . . . . 9 (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = ({𝐶} ∩ ({𝐴} ∪ {𝐵}))
10 indi 3854 . . . . . . . . 9 ({𝐶} ∩ ({𝐴} ∪ {𝐵})) = (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵}))
119, 10eqtri 2648 . . . . . . . 8 (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵}))
1211a1i 11 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵})))
13 necom 2849 . . . . . . . . . . 11 (𝐴𝐶𝐶𝐴)
14 disjsn2 4222 . . . . . . . . . . 11 (𝐶𝐴 → ({𝐶} ∩ {𝐴}) = ∅)
1513, 14sylbi 207 . . . . . . . . . 10 (𝐴𝐶 → ({𝐶} ∩ {𝐴}) = ∅)
1615adantr 481 . . . . . . . . 9 ((𝐴𝐶𝐵𝐶) → ({𝐶} ∩ {𝐴}) = ∅)
17 necom 2849 . . . . . . . . . . 11 (𝐵𝐶𝐶𝐵)
18 disjsn2 4222 . . . . . . . . . . 11 (𝐶𝐵 → ({𝐶} ∩ {𝐵}) = ∅)
1917, 18sylbi 207 . . . . . . . . . 10 (𝐵𝐶 → ({𝐶} ∩ {𝐵}) = ∅)
2019adantl 482 . . . . . . . . 9 ((𝐴𝐶𝐵𝐶) → ({𝐶} ∩ {𝐵}) = ∅)
2116, 20uneq12d 3751 . . . . . . . 8 ((𝐴𝐶𝐵𝐶) → (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵})) = (∅ ∪ ∅))
22 unidm 3739 . . . . . . . 8 (∅ ∪ ∅) = ∅
2321, 22syl6eq 2676 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵})) = ∅)
248, 12, 233eqtrd 2664 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
25 disj3 3998 . . . . . 6 (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶}))
2624, 25sylib 208 . . . . 5 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶}))
2726eqcomd 2632 . . . 4 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∖ {𝐶}) = {𝐴, 𝐵})
28 difid 3927 . . . . 5 ({𝐶} ∖ {𝐶}) = ∅
2928a1i 11 . . . 4 ((𝐴𝐶𝐵𝐶) → ({𝐶} ∖ {𝐶}) = ∅)
3027, 29uneq12d 3751 . . 3 ((𝐴𝐶𝐵𝐶) → (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = ({𝐴, 𝐵} ∪ ∅))
31 un0 3944 . . 3 ({𝐴, 𝐵} ∪ ∅) = {𝐴, 𝐵}
3230, 31syl6eq 2676 . 2 ((𝐴𝐶𝐵𝐶) → (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = {𝐴, 𝐵})
333, 5, 323eqtrd 2664 1 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wne 2796  cdif 3557  cun 3558  cin 3559  c0 3896  {csn 4153  {cpr 4155  {ctp 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-sn 4154  df-pr 4156  df-tp 4158
This theorem is referenced by: (None)
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