Theorem undifdc 6741

Description: Union of complementary parts into whole. This is a case where we can strengthen undifss 3390 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.)

Assertion

Ref	Expression
undifdc	⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem undifdc

Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ (𝑤 = ∅ → 𝑤 = ∅)
2		difeq2 3135	. . . 4 ⊢ (𝑤 = ∅ → (𝐴 ∖ 𝑤) = (𝐴 ∖ ∅))
3	1, 2	uneq12d 3178	. . 3 ⊢ (𝑤 = ∅ → (𝑤 ∪ (𝐴 ∖ 𝑤)) = (∅ ∪ (𝐴 ∖ ∅)))
4	3	eqeq2d 2111	. 2 ⊢ (𝑤 = ∅ → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))))
5		id 19	. . . 4 ⊢ (𝑤 = 𝑣 → 𝑤 = 𝑣)
6		difeq2 3135	. . . 4 ⊢ (𝑤 = 𝑣 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑣))
7	5, 6	uneq12d 3178	. . 3 ⊢ (𝑤 = 𝑣 → (𝑤 ∪ (𝐴 ∖ 𝑤)) = (𝑣 ∪ (𝐴 ∖ 𝑣)))
8	7	eqeq2d 2111	. 2 ⊢ (𝑤 = 𝑣 → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))))
9		id 19	. . . 4 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → 𝑤 = (𝑣 ∪ {𝑧}))
10		difeq2 3135	. . . 4 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 ∖ 𝑤) = (𝐴 ∖ (𝑣 ∪ {𝑧})))
11	9, 10	uneq12d 3178	. . 3 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → (𝑤 ∪ (𝐴 ∖ 𝑤)) = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
12	11	eqeq2d 2111	. 2 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
13		id 19	. . . 4 ⊢ (𝑤 = 𝐵 → 𝑤 = 𝐵)
14		difeq2 3135	. . . 4 ⊢ (𝑤 = 𝐵 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝐵))
15	13, 14	uneq12d 3178	. . 3 ⊢ (𝑤 = 𝐵 → (𝑤 ∪ (𝐴 ∖ 𝑤)) = (𝐵 ∪ (𝐴 ∖ 𝐵)))
16	15	eqeq2d 2111	. 2 ⊢ (𝑤 = 𝐵 → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))))
17		un0 3343	. . . 4 ⊢ ((𝐴 ∖ ∅) ∪ ∅) = (𝐴 ∖ ∅)
18		uncom 3167	. . . 4 ⊢ ((𝐴 ∖ ∅) ∪ ∅) = (∅ ∪ (𝐴 ∖ ∅))
19		dif0 3380	. . . 4 ⊢ (𝐴 ∖ ∅) = 𝐴
20	17, 18, 19	3eqtr3ri 2129	. . 3 ⊢ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))
21	20	a1i 9	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (∅ ∪ (𝐴 ∖ ∅)))
22		difundi 3275	. . . . . . 7 ⊢ (𝐴 ∖ (𝑣 ∪ {𝑧})) = ((𝐴 ∖ 𝑣) ∩ (𝐴 ∖ {𝑧}))
23	22	uneq2i 3174	. . . . . 6 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = ((𝑣 ∪ {𝑧}) ∪ ((𝐴 ∖ 𝑣) ∩ (𝐴 ∖ {𝑧})))
24		undi 3271	. . . . . 6 ⊢ ((𝑣 ∪ {𝑧}) ∪ ((𝐴 ∖ 𝑣) ∩ (𝐴 ∖ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
25	23, 24	eqtri 2120	. . . . 5 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
26		simp3 951	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
27	26	ad3antrrr 479	. . . . . . . . . . 11 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → 𝐵 ⊆ 𝐴)
28		simplrr 506	. . . . . . . . . . . 12 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → 𝑧 ∈ (𝐵 ∖ 𝑣))
29	28	eldifad 3032	. . . . . . . . . . 11 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → 𝑧 ∈ 𝐵)
30	27, 29	sseldd 3048	. . . . . . . . . 10 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → 𝑧 ∈ 𝐴)
31	30	snssd 3612	. . . . . . . . 9 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → {𝑧} ⊆ 𝐴)
32		ssequn1 3193	. . . . . . . . 9 ⊢ ({𝑧} ⊆ 𝐴 ↔ ({𝑧} ∪ 𝐴) = 𝐴)
33	31, 32	sylib 121	. . . . . . . 8 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → ({𝑧} ∪ 𝐴) = 𝐴)
34		simpr 109	. . . . . . . . 9 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣)))
35	34	uneq2d 3177	. . . . . . . 8 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → ({𝑧} ∪ 𝐴) = ({𝑧} ∪ (𝑣 ∪ (𝐴 ∖ 𝑣))))
36	33, 35	eqtr3d 2134	. . . . . . 7 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → 𝐴 = ({𝑧} ∪ (𝑣 ∪ (𝐴 ∖ 𝑣))))
37		uncom 3167	. . . . . . . . 9 ⊢ (𝑣 ∪ {𝑧}) = ({𝑧} ∪ 𝑣)
38	37	uneq1i 3173	. . . . . . . 8 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) = (({𝑧} ∪ 𝑣) ∪ (𝐴 ∖ 𝑣))
39		unass 3180	. . . . . . . 8 ⊢ (({𝑧} ∪ 𝑣) ∪ (𝐴 ∖ 𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴 ∖ 𝑣)))
40	38, 39	eqtri 2120	. . . . . . 7 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴 ∖ 𝑣)))
41	36, 40	syl6reqr 2151	. . . . . 6 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) = 𝐴)
42		unass 3180	. . . . . . . 8 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧})))
43		uncom 3167	. . . . . . . . . 10 ⊢ ({𝑧} ∪ (𝐴 ∖ {𝑧})) = ((𝐴 ∖ {𝑧}) ∪ {𝑧})
44		simp1 949	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
45	44	ad3antrrr 479	. . . . . . . . . . 11 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
46		dcdifsnid 6330	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑧 ∈ 𝐴) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
47	45, 30, 46	syl2anc 406	. . . . . . . . . 10 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
48	43, 47	syl5eq 2144	. . . . . . . . 9 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → ({𝑧} ∪ (𝐴 ∖ {𝑧})) = 𝐴)
49	48	uneq2d 3177	. . . . . . . 8 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧}))) = (𝑣 ∪ 𝐴))
50	42, 49	syl5eq 2144	. . . . . . 7 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣 ∪ 𝐴))
51		simplrl 505	. . . . . . . . 9 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → 𝑣 ⊆ 𝐵)
52	51, 27	sstrd 3057	. . . . . . . 8 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → 𝑣 ⊆ 𝐴)
53		ssequn1 3193	. . . . . . . 8 ⊢ (𝑣 ⊆ 𝐴 ↔ (𝑣 ∪ 𝐴) = 𝐴)
54	52, 53	sylib 121	. . . . . . 7 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → (𝑣 ∪ 𝐴) = 𝐴)
55	50, 54	eqtrd 2132	. . . . . 6 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = 𝐴)
56	41, 55	ineq12d 3225	. . . . 5 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → (((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧}))) = (𝐴 ∩ 𝐴))
57	25, 56	syl5eq 2144	. . . 4 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (𝐴 ∩ 𝐴))
58		inidm 3232	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
59	57, 58	syl6req 2149	. . 3 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))) → 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
60	59	ex 114	. 2 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑣))) → (𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣)) → 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
61		simp2 950	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
62	4, 8, 12, 16, 21, 60, 61	findcard2sd 6715	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 786   ∧ w3a 930   = wceq 1299   ∈ wcel 1448  ∀wral 2375   ∖ cdif 3018   ∪ cun 3019   ∩ cin 3020   ⊆ wss 3021  ∅c0 3310  {csn 3474  Fincfn 6564

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-er 6359  df-en 6565  df-fin 6567

This theorem is referenced by:  undiffi  6742