Theorem eluni1g 4172

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

Assertion

Ref	Expression
eluni1g	|- (A e. V -> (A e. ⋃1B <-> {A} e. 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 = U.(B i^i 1c)
2	1	eleq2i 2417	. . 3 |- (A e. ⋃1B <-> A e. U.(B i^i 1c))
3		eluni 3894	. . 3 |- (A e. U.(B i^i 1c) <-> E.x(A e. x /\ x e. (B i^i 1c)))
4		elin 3219	. . . . . . . 8 |- (x e. (B i^i 1c) <-> (x e. B /\ x e. 1c))
5		ancom 437	. . . . . . . 8 |- ((x e. B /\ x e. 1c) <-> (x e. 1c /\ x e. B))
6		el1c 4139	. . . . . . . . . 10 |- (x e. 1c <-> E.y x = {y})
7	6	anbi1i 676	. . . . . . . . 9 |- ((x e. 1c /\ x e. B) <-> (E.y x = {y} /\ x e. B))
8		19.41v 1901	. . . . . . . . 9 |- (E.y(x = {y} /\ x e. B) <-> (E.y x = {y} /\ x e. B))
9	7, 8	bitr4i 243	. . . . . . . 8 |- ((x e. 1c /\ x e. B) <-> E.y(x = {y} /\ x e. B))
10	4, 5, 9	3bitri 262	. . . . . . 7 |- (x e. (B i^i 1c) <-> E.y(x = {y} /\ x e. B))
11	10	anbi2i 675	. . . . . 6 |- ((A e. x /\ x e. (B i^i 1c)) <-> (A e. x /\ E.y(x = {y} /\ x e. B)))
12		19.42v 1905	. . . . . 6 |- (E.y(A e. x /\ (x = {y} /\ x e. B)) <-> (A e. x /\ E.y(x = {y} /\ x e. B)))
13	11, 12	bitr4i 243	. . . . 5 |- ((A e. x /\ x e. (B i^i 1c)) <-> E.y(A e. x /\ (x = {y} /\ x e. B)))
14	13	exbii 1582	. . . 4 |- (E.x(A e. x /\ x e. (B i^i 1c)) <-> E.xE.y(A e. x /\ (x = {y} /\ x e. B)))
15		excom 1741	. . . 4 |- (E.xE.y(A e. x /\ (x = {y} /\ x e. B)) <-> E.yE.x(A e. x /\ (x = {y} /\ x e. B)))
16		an12 772	. . . . . . 7 |- ((A e. x /\ (x = {y} /\ x e. B)) <-> (x = {y} /\ (A e. x /\ x e. B)))
17	16	exbii 1582	. . . . . 6 |- (E.x(A e. x /\ (x = {y} /\ x e. B)) <-> E.x(x = {y} /\ (A e. x /\ x e. B)))
18		snex 4111	. . . . . . 7 |- {y} e. _V
19		eleq2 2414	. . . . . . . . 9 |- (x = {y} -> (A e. x <-> A e. {y}))
20		vex 2862	. . . . . . . . . 10 |- y e. _V
21	20	elsnc2 3762	. . . . . . . . 9 |- (A e. {y} <-> A = y)
22	19, 21	syl6bb 252	. . . . . . . 8 |- (x = {y} -> (A e. x <-> A = y))
23		eleq1 2413	. . . . . . . 8 |- (x = {y} -> (x e. B <-> {y} e. B))
24	22, 23	anbi12d 691	. . . . . . 7 |- (x = {y} -> ((A e. x /\ x e. B) <-> (A = y /\ {y} e. B)))
25	18, 24	ceqsexv 2894	. . . . . 6 |- (E.x(x = {y} /\ (A e. x /\ x e. B)) <-> (A = y /\ {y} e. B))
26		eqcom 2355	. . . . . . 7 |- (A = y <-> y = A)
27	26	anbi1i 676	. . . . . 6 |- ((A = y /\ {y} e. B) <-> (y = A /\ {y} e. B))
28	17, 25, 27	3bitri 262	. . . . 5 |- (E.x(A e. x /\ (x = {y} /\ x e. B)) <-> (y = A /\ {y} e. B))
29	28	exbii 1582	. . . 4 |- (E.yE.x(A e. x /\ (x = {y} /\ x e. B)) <-> E.y(y = A /\ {y} e. B))
30	14, 15, 29	3bitri 262	. . 3 |- (E.x(A e. x /\ x e. (B i^i 1c)) <-> E.y(y = A /\ {y} e. B))
31	2, 3, 30	3bitri 262	. 2 |- (A e. ⋃1B <-> E.y(y = A /\ {y} e. B))
32		sneq 3744	. . . 4 |- (y = A -> {y} = {A})
33	32	eleq1d 2419	. . 3 |- (y = A -> ({y} e. B <-> {A} e. B))
34	33	ceqsexgv 2971	. 2 |- (A e. V -> (E.y(y = A /\ {y} e. B) <-> {A} e. B))
35	31, 34	syl5bb 248	1 |- (A e. V -> (A e. ⋃1B <-> {A} e. B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   i^i cin 3208  {csn 3737  U.cuni 3891  ⋃₁cuni1 4133  1_cc1c 4134

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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