Theorem diftpsn3 3850

Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Assertion

Ref	Expression
diftpsn3	⊢ ((A ≠ C ∧ B ≠ C) → ({A, B, C} ∖ {C}) = {A, B})

Proof of Theorem diftpsn3

Step	Hyp	Ref	Expression
1		df-tp 3744	. . . 4 ⊢ {A, B, C} = ({A, B} ∪ {C})
2	1	a1i 10	. . 3 ⊢ ((A ≠ C ∧ B ≠ C) → {A, B, C} = ({A, B} ∪ {C}))
3	2	difeq1d 3385	. 2 ⊢ ((A ≠ C ∧ B ≠ C) → ({A, B, C} ∖ {C}) = (({A, B} ∪ {C}) ∖ {C}))
4		difundir 3509	. . 3 ⊢ (({A, B} ∪ {C}) ∖ {C}) = (({A, B} ∖ {C}) ∪ ({C} ∖ {C}))
5	4	a1i 10	. 2 ⊢ ((A ≠ C ∧ B ≠ C) → (({A, B} ∪ {C}) ∖ {C}) = (({A, B} ∖ {C}) ∪ ({C} ∖ {C})))
6		df-pr 3743	. . . . . . . . 9 ⊢ {A, B} = ({A} ∪ {B})
7	6	a1i 10	. . . . . . . 8 ⊢ ((A ≠ C ∧ B ≠ C) → {A, B} = ({A} ∪ {B}))
8	7	ineq1d 3457	. . . . . . 7 ⊢ ((A ≠ C ∧ B ≠ C) → ({A, B} ∩ {C}) = (({A} ∪ {B}) ∩ {C}))
9		incom 3449	. . . . . . . . 9 ⊢ (({A} ∪ {B}) ∩ {C}) = ({C} ∩ ({A} ∪ {B}))
10		indi 3502	. . . . . . . . 9 ⊢ ({C} ∩ ({A} ∪ {B})) = (({C} ∩ {A}) ∪ ({C} ∩ {B}))
11	9, 10	eqtri 2373	. . . . . . . 8 ⊢ (({A} ∪ {B}) ∩ {C}) = (({C} ∩ {A}) ∪ ({C} ∩ {B}))
12	11	a1i 10	. . . . . . 7 ⊢ ((A ≠ C ∧ B ≠ C) → (({A} ∪ {B}) ∩ {C}) = (({C} ∩ {A}) ∪ ({C} ∩ {B})))
13		necom 2598	. . . . . . . . . . 11 ⊢ (A ≠ C ↔ C ≠ A)
14		disjsn2 3788	. . . . . . . . . . 11 ⊢ (C ≠ A → ({C} ∩ {A}) = ∅)
15	13, 14	sylbi 187	. . . . . . . . . 10 ⊢ (A ≠ C → ({C} ∩ {A}) = ∅)
16	15	adantr 451	. . . . . . . . 9 ⊢ ((A ≠ C ∧ B ≠ C) → ({C} ∩ {A}) = ∅)
17		necom 2598	. . . . . . . . . . 11 ⊢ (B ≠ C ↔ C ≠ B)
18		disjsn2 3788	. . . . . . . . . . 11 ⊢ (C ≠ B → ({C} ∩ {B}) = ∅)
19	17, 18	sylbi 187	. . . . . . . . . 10 ⊢ (B ≠ C → ({C} ∩ {B}) = ∅)
20	19	adantl 452	. . . . . . . . 9 ⊢ ((A ≠ C ∧ B ≠ C) → ({C} ∩ {B}) = ∅)
21	16, 20	uneq12d 3420	. . . . . . . 8 ⊢ ((A ≠ C ∧ B ≠ C) → (({C} ∩ {A}) ∪ ({C} ∩ {B})) = (∅ ∪ ∅))
22		unidm 3408	. . . . . . . 8 ⊢ (∅ ∪ ∅) = ∅
23	21, 22	syl6eq 2401	. . . . . . 7 ⊢ ((A ≠ C ∧ B ≠ C) → (({C} ∩ {A}) ∪ ({C} ∩ {B})) = ∅)
24	8, 12, 23	3eqtrd 2389	. . . . . 6 ⊢ ((A ≠ C ∧ B ≠ C) → ({A, B} ∩ {C}) = ∅)
25		disj3 3596	. . . . . 6 ⊢ (({A, B} ∩ {C}) = ∅ ↔ {A, B} = ({A, B} ∖ {C}))
26	24, 25	sylib 188	. . . . 5 ⊢ ((A ≠ C ∧ B ≠ C) → {A, B} = ({A, B} ∖ {C}))
27	26	eqcomd 2358	. . . 4 ⊢ ((A ≠ C ∧ B ≠ C) → ({A, B} ∖ {C}) = {A, B})
28		difid 3619	. . . . 5 ⊢ ({C} ∖ {C}) = ∅
29	28	a1i 10	. . . 4 ⊢ ((A ≠ C ∧ B ≠ C) → ({C} ∖ {C}) = ∅)
30	27, 29	uneq12d 3420	. . 3 ⊢ ((A ≠ C ∧ B ≠ C) → (({A, B} ∖ {C}) ∪ ({C} ∖ {C})) = ({A, B} ∪ ∅))
31		un0 3576	. . 3 ⊢ ({A, B} ∪ ∅) = {A, B}
32	30, 31	syl6eq 2401	. 2 ⊢ ((A ≠ C ∧ B ≠ C) → (({A, B} ∖ {C}) ∪ ({C} ∖ {C})) = {A, B})
33	3, 5, 32	3eqtrd 2389	1 ⊢ ((A ≠ C ∧ B ≠ C) → ({A, B, C} ∖ {C}) = {A, B})

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2517   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  {cpr 3739  {ctp 3740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-tp 3744

This theorem is referenced by: (None)