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Theorem diftpsn3OLD 4365
Description: Obsolete proof of diftpsn3 4364 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 4215 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
21a1i 11 . . 3 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}))
32difeq1d 3760 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}))
4 difundir 3913 . . 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 4213 . . . . . . . . 9 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
76a1i 11 . . . . . . . 8 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}))
87ineq1d 3846 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = (({𝐴} ∪ {𝐵}) ∩ {𝐶}))
9 incom 3838 . . . . . . . . 9 (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = ({𝐶} ∩ ({𝐴} ∪ {𝐵}))
10 indi 3906 . . . . . . . . 9 ({𝐶} ∩ ({𝐴} ∪ {𝐵})) = (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵}))
119, 10eqtri 2673 . . . . . . . 8 (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵}))
1211a1i 11 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → (({𝐴} ∪ {𝐵}) ∩ {𝐶}) = (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵})))
13 necom 2876 . . . . . . . . . . 11 (𝐴𝐶𝐶𝐴)
14 disjsn2 4279 . . . . . . . . . . 11 (𝐶𝐴 → ({𝐶} ∩ {𝐴}) = ∅)
1513, 14sylbi 207 . . . . . . . . . 10 (𝐴𝐶 → ({𝐶} ∩ {𝐴}) = ∅)
1615adantr 480 . . . . . . . . 9 ((𝐴𝐶𝐵𝐶) → ({𝐶} ∩ {𝐴}) = ∅)
17 necom 2876 . . . . . . . . . . 11 (𝐵𝐶𝐶𝐵)
18 disjsn2 4279 . . . . . . . . . . 11 (𝐶𝐵 → ({𝐶} ∩ {𝐵}) = ∅)
1917, 18sylbi 207 . . . . . . . . . 10 (𝐵𝐶 → ({𝐶} ∩ {𝐵}) = ∅)
2019adantl 481 . . . . . . . . 9 ((𝐴𝐶𝐵𝐶) → ({𝐶} ∩ {𝐵}) = ∅)
2116, 20uneq12d 3801 . . . . . . . 8 ((𝐴𝐶𝐵𝐶) → (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵})) = (∅ ∪ ∅))
22 unidm 3789 . . . . . . . 8 (∅ ∪ ∅) = ∅
2321, 22syl6eq 2701 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → (({𝐶} ∩ {𝐴}) ∪ ({𝐶} ∩ {𝐵})) = ∅)
248, 12, 233eqtrd 2689 . . . . . 6 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
25 disj3 4054 . . . . . 6 (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶}))
2624, 25sylib 208 . . . . 5 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶}))
2726eqcomd 2657 . . . 4 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∖ {𝐶}) = {𝐴, 𝐵})
28 difid 3981 . . . . 5 ({𝐶} ∖ {𝐶}) = ∅
2928a1i 11 . . . 4 ((𝐴𝐶𝐵𝐶) → ({𝐶} ∖ {𝐶}) = ∅)
3027, 29uneq12d 3801 . . 3 ((𝐴𝐶𝐵𝐶) → (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = ({𝐴, 𝐵} ∪ ∅))
31 un0 4000 . . 3 ({𝐴, 𝐵} ∪ ∅) = {𝐴, 𝐵}
3230, 31syl6eq 2701 . 2 ((𝐴𝐶𝐵𝐶) → (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = {𝐴, 𝐵})
333, 5, 323eqtrd 2689 1 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wne 2823  cdif 3604  cun 3605  cin 3606  c0 3948  {csn 4210  {cpr 4212  {ctp 4214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-tp 4215
This theorem is referenced by: (None)
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