Theorem eluni1g 4172

Description: Membership in a unit union. (Contributed by SF, 15-Mar-2015.)

Assertion

Ref	Expression
eluni1g	⊢ (A ∈ V → (A ∈ ⋃1B ↔ {A} ∈ B))

Proof of Theorem eluni1g

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-uni1 4138	. . . 4 ⊢ ⋃1B = ∪(B ∩ 1c)
2	1	eleq2i 2417	. . 3 ⊢ (A ∈ ⋃1B ↔ A ∈ ∪(B ∩ 1c))
3		eluni 3894	. . 3 ⊢ (A ∈ ∪(B ∩ 1c) ↔ ∃x(A ∈ x ∧ x ∈ (B ∩ 1c)))
4		elin 3219	. . . . . . . 8 ⊢ (x ∈ (B ∩ 1c) ↔ (x ∈ B ∧ x ∈ 1c))
5		ancom 437	. . . . . . . 8 ⊢ ((x ∈ B ∧ x ∈ 1c) ↔ (x ∈ 1c ∧ x ∈ B))
6		el1c 4139	. . . . . . . . . 10 ⊢ (x ∈ 1c ↔ ∃y x = {y})
7	6	anbi1i 676	. . . . . . . . 9 ⊢ ((x ∈ 1c ∧ x ∈ B) ↔ (∃y x = {y} ∧ x ∈ B))
8		19.41v 1901	. . . . . . . . 9 ⊢ (∃y(x = {y} ∧ x ∈ B) ↔ (∃y x = {y} ∧ x ∈ B))
9	7, 8	bitr4i 243	. . . . . . . 8 ⊢ ((x ∈ 1c ∧ x ∈ B) ↔ ∃y(x = {y} ∧ x ∈ B))
10	4, 5, 9	3bitri 262	. . . . . . 7 ⊢ (x ∈ (B ∩ 1c) ↔ ∃y(x = {y} ∧ x ∈ B))
11	10	anbi2i 675	. . . . . 6 ⊢ ((A ∈ x ∧ x ∈ (B ∩ 1c)) ↔ (A ∈ x ∧ ∃y(x = {y} ∧ x ∈ B)))
12		19.42v 1905	. . . . . 6 ⊢ (∃y(A ∈ x ∧ (x = {y} ∧ x ∈ B)) ↔ (A ∈ x ∧ ∃y(x = {y} ∧ x ∈ B)))
13	11, 12	bitr4i 243	. . . . 5 ⊢ ((A ∈ x ∧ x ∈ (B ∩ 1c)) ↔ ∃y(A ∈ x ∧ (x = {y} ∧ x ∈ B)))
14	13	exbii 1582	. . . 4 ⊢ (∃x(A ∈ x ∧ x ∈ (B ∩ 1c)) ↔ ∃x∃y(A ∈ x ∧ (x = {y} ∧ x ∈ B)))
15		excom 1741	. . . 4 ⊢ (∃x∃y(A ∈ x ∧ (x = {y} ∧ x ∈ B)) ↔ ∃y∃x(A ∈ x ∧ (x = {y} ∧ x ∈ B)))
16		an12 772	. . . . . . 7 ⊢ ((A ∈ x ∧ (x = {y} ∧ x ∈ B)) ↔ (x = {y} ∧ (A ∈ x ∧ x ∈ B)))
17	16	exbii 1582	. . . . . 6 ⊢ (∃x(A ∈ x ∧ (x = {y} ∧ x ∈ B)) ↔ ∃x(x = {y} ∧ (A ∈ x ∧ x ∈ B)))
18		snex 4111	. . . . . . 7 ⊢ {y} ∈ V
19		eleq2 2414	. . . . . . . . 9 ⊢ (x = {y} → (A ∈ x ↔ A ∈ {y}))
20		vex 2862	. . . . . . . . . 10 ⊢ y ∈ V
21	20	elsnc2 3762	. . . . . . . . 9 ⊢ (A ∈ {y} ↔ A = y)
22	19, 21	syl6bb 252	. . . . . . . 8 ⊢ (x = {y} → (A ∈ x ↔ A = y))
23		eleq1 2413	. . . . . . . 8 ⊢ (x = {y} → (x ∈ B ↔ {y} ∈ B))
24	22, 23	anbi12d 691	. . . . . . 7 ⊢ (x = {y} → ((A ∈ x ∧ x ∈ B) ↔ (A = y ∧ {y} ∈ B)))
25	18, 24	ceqsexv 2894	. . . . . 6 ⊢ (∃x(x = {y} ∧ (A ∈ x ∧ x ∈ B)) ↔ (A = y ∧ {y} ∈ B))
26		eqcom 2355	. . . . . . 7 ⊢ (A = y ↔ y = A)
27	26	anbi1i 676	. . . . . 6 ⊢ ((A = y ∧ {y} ∈ B) ↔ (y = A ∧ {y} ∈ B))
28	17, 25, 27	3bitri 262	. . . . 5 ⊢ (∃x(A ∈ x ∧ (x = {y} ∧ x ∈ B)) ↔ (y = A ∧ {y} ∈ B))
29	28	exbii 1582	. . . 4 ⊢ (∃y∃x(A ∈ x ∧ (x = {y} ∧ x ∈ B)) ↔ ∃y(y = A ∧ {y} ∈ B))
30	14, 15, 29	3bitri 262	. . 3 ⊢ (∃x(A ∈ x ∧ x ∈ (B ∩ 1c)) ↔ ∃y(y = A ∧ {y} ∈ B))
31	2, 3, 30	3bitri 262	. 2 ⊢ (A ∈ ⋃1B ↔ ∃y(y = A ∧ {y} ∈ B))
32		sneq 3744	. . . 4 ⊢ (y = A → {y} = {A})
33	32	eleq1d 2419	. . 3 ⊢ (y = A → ({y} ∈ B ↔ {A} ∈ B))
34	33	ceqsexgv 2971	. 2 ⊢ (A ∈ V → (∃y(y = A ∧ {y} ∈ B) ↔ {A} ∈ B))
35	31, 34	syl5bb 248	1 ⊢ (A ∈ V → (A ∈ ⋃1B ↔ {A} ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  {csn 3737  ∪cuni 3891  ⋃₁cuni1 4133  1_cc1c 4134

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  ax-nin 4078  ax-sn 4087

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-uni 3892  df-1c 4136  df-uni1 4138

This theorem is referenced by:  eluni1  4173