Theorem undifdc 6953

Description: Union of complementary parts into whole. This is a case where we can strengthen undifss 3518 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 3262	. . . 4 ⊢ (𝑤 = ∅ → (𝐴 ∖ 𝑤) = (𝐴 ∖ ∅))
3	1, 2	uneq12d 3305	. . 3 ⊢ (𝑤 = ∅ → (𝑤 ∪ (𝐴 ∖ 𝑤)) = (∅ ∪ (𝐴 ∖ ∅)))
4	3	eqeq2d 2201	. 2 ⊢ (𝑤 = ∅ → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))))
5		id 19	. . . 4 ⊢ (𝑤 = 𝑣 → 𝑤 = 𝑣)
6		difeq2 3262	. . . 4 ⊢ (𝑤 = 𝑣 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑣))
7	5, 6	uneq12d 3305	. . 3 ⊢ (𝑤 = 𝑣 → (𝑤 ∪ (𝐴 ∖ 𝑤)) = (𝑣 ∪ (𝐴 ∖ 𝑣)))
8	7	eqeq2d 2201	. 2 ⊢ (𝑤 = 𝑣 → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = (𝑣 ∪ (𝐴 ∖ 𝑣))))
9		id 19	. . . 4 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → 𝑤 = (𝑣 ∪ {𝑧}))
10		difeq2 3262	. . . 4 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 ∖ 𝑤) = (𝐴 ∖ (𝑣 ∪ {𝑧})))
11	9, 10	uneq12d 3305	. . 3 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → (𝑤 ∪ (𝐴 ∖ 𝑤)) = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
12	11	eqeq2d 2201	. 2 ⊢ (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
13		id 19	. . . 4 ⊢ (𝑤 = 𝐵 → 𝑤 = 𝐵)
14		difeq2 3262	. . . 4 ⊢ (𝑤 = 𝐵 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝐵))
15	13, 14	uneq12d 3305	. . 3 ⊢ (𝑤 = 𝐵 → (𝑤 ∪ (𝐴 ∖ 𝑤)) = (𝐵 ∪ (𝐴 ∖ 𝐵)))
16	15	eqeq2d 2201	. 2 ⊢ (𝑤 = 𝐵 → (𝐴 = (𝑤 ∪ (𝐴 ∖ 𝑤)) ↔ 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))))
17		un0 3471	. . . 4 ⊢ ((𝐴 ∖ ∅) ∪ ∅) = (𝐴 ∖ ∅)
18		uncom 3294	. . . 4 ⊢ ((𝐴 ∖ ∅) ∪ ∅) = (∅ ∪ (𝐴 ∖ ∅))
19		dif0 3508	. . . 4 ⊢ (𝐴 ∖ ∅) = 𝐴
20	17, 18, 19	3eqtr3ri 2219	. . 3 ⊢ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))
21	20	a1i 9	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (∅ ∪ (𝐴 ∖ ∅)))
22		difundi 3402	. . . . . . 7 ⊢ (𝐴 ∖ (𝑣 ∪ {𝑧})) = ((𝐴 ∖ 𝑣) ∩ (𝐴 ∖ {𝑧}))
23	22	uneq2i 3301	. . . . . 6 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = ((𝑣 ∪ {𝑧}) ∪ ((𝐴 ∖ 𝑣) ∩ (𝐴 ∖ {𝑧})))
24		undi 3398	. . . . . 6 ⊢ ((𝑣 ∪ {𝑧}) ∪ ((𝐴 ∖ 𝑣) ∩ (𝐴 ∖ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
25	23, 24	eqtri 2210	. . . . 5 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
26		uncom 3294	. . . . . . . . 9 ⊢ (𝑣 ∪ {𝑧}) = ({𝑧} ∪ 𝑣)
27	26	uneq1i 3300	. . . . . . . 8 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) = (({𝑧} ∪ 𝑣) ∪ (𝐴 ∖ 𝑣))
28		unass 3307	. . . . . . . 8 ⊢ (({𝑧} ∪ 𝑣) ∪ (𝐴 ∖ 𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴 ∖ 𝑣)))
29	27, 28	eqtri 2210	. . . . . . 7 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ 𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴 ∖ 𝑣)))
30		simp3 1001	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
31	30	ad3antrrr 492	. . . . . . . . . . 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 536	. . . . . . . . . . . 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 3155	. . . . . . . . . . 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 3171	. . . . . . . . . 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 3752	. . . . . . . . 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 3320	. . . . . . . . 9 ⊢ ({𝑧} ⊆ 𝐴 ↔ ({𝑧} ∪ 𝐴) = 𝐴)
37	35, 36	sylib 122	. . . . . . . 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 110	. . . . . . . . 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 3304	. . . . . . . 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 2224	. . . . . . 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 2241	. . . . . 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 3307	. . . . . . . 8 ⊢ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧})))
43		uncom 3294	. . . . . . . . . 10 ⊢ ({𝑧} ∪ (𝐴 ∖ {𝑧})) = ((𝐴 ∖ {𝑧}) ∪ {𝑧})
44		simp1 999	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
45	44	ad3antrrr 492	. . . . . . . . . . 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 6530	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑧 ∈ 𝐴) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
47	45, 34, 46	syl2anc 411	. . . . . . . . . 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	eqtrid 2234	. . . . . . . . 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 3304	. . . . . . . 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	eqtrid 2234	. . . . . . 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 535	. . . . . . . . 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 3180	. . . . . . . 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 3320	. . . . . . . 8 ⊢ (𝑣 ⊆ 𝐴 ↔ (𝑣 ∪ 𝐴) = 𝐴)
54	52, 53	sylib 122	. . . . . . 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 2222	. . . . . 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 3352	. . . . 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	eqtrid 2234	. . . 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 3359	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
59	57, 58	eqtr2di 2239	. . 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 115	. 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 1000	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
62	4, 8, 12, 16, 21, 60, 61	findcard2sd 6921	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  ∀wral 2468   ∖ cdif 3141   ∪ cun 3142   ∩ cin 3143   ⊆ wss 3144  ∅c0 3437  {csn 3607  Fincfn 6767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-er 6560  df-en 6768  df-fin 6770

This theorem is referenced by:  undiffi  6954