Theorem undifdc 6889

Description: Union of complementary parts into whole. This is a case where we can strengthen undifss 3489 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 3234	. . . 4 ⊢ (𝑤 = ∅ → (𝐴 ∖ 𝑤) = (𝐴 ∖ ∅))
3	1, 2	uneq12d 3277	. . 3 ⊢ (𝑤 = ∅ → (𝑤 ∪ (𝐴 ∖ 𝑤)) = (∅ ∪ (𝐴 ∖ ∅)))
4	3	eqeq2d 2177	. 2 ⊢ (𝑤 = ∅ → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))))
5		id 19	. . . 4 ⊢ (𝑤 = 𝑣 → 𝑤 = 𝑣)
6		difeq2 3234	. . . 4 ⊢ (𝑤 = 𝑣 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑣))
7	5, 6	uneq12d 3277	. . 3 ⊢ (𝑤 = 𝑣 → (𝑤 ∪ (𝐴 ∖ 𝑤)) = (𝑣 ∪ (𝐴 ∖ 𝑣)))
8	7	eqeq2d 2177	. 2 ⊢ (𝑤 = 𝑣 → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))))
9		id 19	. . . 4 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → 𝑤 = (𝑣 ∪ {𝑧}))
10		difeq2 3234	. . . 4 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 ∖ 𝑤) = (𝐴 ∖ (𝑣 ∪ {𝑧})))
11	9, 10	uneq12d 3277	. . 3 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → (𝑤 ∪ (𝐴 ∖ 𝑤)) = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
12	11	eqeq2d 2177	. 2 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
13		id 19	. . . 4 ⊢ (𝑤 = 𝐵 → 𝑤 = 𝐵)
14		difeq2 3234	. . . 4 ⊢ (𝑤 = 𝐵 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝐵))
15	13, 14	uneq12d 3277	. . 3 ⊢ (𝑤 = 𝐵 → (𝑤 ∪ (𝐴 ∖ 𝑤)) = (𝐵 ∪ (𝐴 ∖ 𝐵)))
16	15	eqeq2d 2177	. 2 ⊢ (𝑤 = 𝐵 → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))))
17		un0 3442	. . . 4 ⊢ ((𝐴 ∖ ∅) ∪ ∅) = (𝐴 ∖ ∅)
18		uncom 3266	. . . 4 ⊢ ((𝐴 ∖ ∅) ∪ ∅) = (∅ ∪ (𝐴 ∖ ∅))
19		dif0 3479	. . . 4 ⊢ (𝐴 ∖ ∅) = 𝐴
20	17, 18, 19	3eqtr3ri 2195	. . 3 ⊢ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))
21	20	a1i 9	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (∅ ∪ (𝐴 ∖ ∅)))
22		difundi 3374	. . . . . . 7 ⊢ (𝐴 ∖ (𝑣 ∪ {𝑧})) = ((𝐴 ∖ 𝑣) ∩ (𝐴 ∖ {𝑧}))
23	22	uneq2i 3273	. . . . . 6 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = ((𝑣 ∪ {𝑧}) ∪ ((𝐴 ∖ 𝑣) ∩ (𝐴 ∖ {𝑧})))
24		undi 3370	. . . . . 6 ⊢ ((𝑣 ∪ {𝑧}) ∪ ((𝐴 ∖ 𝑣) ∩ (𝐴 ∖ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
25	23, 24	eqtri 2186	. . . . 5 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
26		uncom 3266	. . . . . . . . 9 ⊢ (𝑣 ∪ {𝑧}) = ({𝑧} ∪ 𝑣)
27	26	uneq1i 3272	. . . . . . . 8 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) = (({𝑧} ∪ 𝑣) ∪ (𝐴 ∖ 𝑣))
28		unass 3279	. . . . . . . 8 ⊢ (({𝑧} ∪ 𝑣) ∪ (𝐴 ∖ 𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴 ∖ 𝑣)))
29	27, 28	eqtri 2186	. . . . . . 7 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴 ∖ 𝑣)))
30		simp3 989	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
31	30	ad3antrrr 484	. . . . . . . . . . 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)
32		simplrr 526	. . . . . . . . . . . 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))
33	32	eldifad 3127	. . . . . . . . . . 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)
34	31, 33	sseldd 3143	. . . . . . . . . 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)
35	34	snssd 3718	. . . . . . . . 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)
36		ssequn1 3292	. . . . . . . . 9 ⊢ ({𝑧} ⊆ 𝐴 ↔ ({𝑧} ∪ 𝐴) = 𝐴)
37	35, 36	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)
38		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)))
39	38	uneq2d 3276	. . . . . . . 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))))
40	37, 39	eqtr3d 2200	. . . . . . 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))))
41	29, 40	eqtr4id 2218	. . . . . 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 3279	. . . . . . . 8 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧})))
43		uncom 3266	. . . . . . . . . 10 ⊢ ({𝑧} ∪ (𝐴 ∖ {𝑧})) = ((𝐴 ∖ {𝑧}) ∪ {𝑧})
44		simp1 987	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
45	44	ad3antrrr 484	. . . . . . . . . . 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 6472	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑧 ∈ 𝐴) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
47	45, 34, 46	syl2anc 409	. . . . . . . . . 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 2211	. . . . . . . . 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 3276	. . . . . . . 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 2211	. . . . . . 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 525	. . . . . . . . 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, 31	sstrd 3152	. . . . . . . 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 3292	. . . . . . . 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 2198	. . . . . 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 3324	. . . . 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 2211	. . . 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 3331	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
59	57, 58	eqtr2di 2216	. . 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 988	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
62	4, 8, 12, 16, 21, 60, 61	findcard2sd 6858	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 824   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∀wral 2444   ∖ cdif 3113   ∪ cun 3114   ∩ cin 3115   ⊆ wss 3116  ∅c0 3409  {csn 3576  Fincfn 6706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-er 6501  df-en 6707  df-fin 6709

This theorem is referenced by:  undiffi  6890