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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.
StepHypRef Expression
1 df-uni1 4138 . . . 4 1B = (B ∩ 1c)
21eleq2i 2417 . . 3 (A 1BA (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 1cy x = {y})
76anbi1i 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))
97, 8bitr4i 243 . . . . . . . 8 ((x 1c x B) ↔ y(x = {y} x B))
104, 5, 93bitri 262 . . . . . . 7 (x (B ∩ 1c) ↔ y(x = {y} x B))
1110anbi2i 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)))
1311, 12bitr4i 243 . . . . 5 ((A x x (B ∩ 1c)) ↔ y(A x (x = {y} x B)))
1413exbii 1582 . . . 4 (x(A x x (B ∩ 1c)) ↔ xy(A x (x = {y} x B)))
15 excom 1741 . . . 4 (xy(A x (x = {y} x B)) ↔ yx(A x (x = {y} x B)))
16 an12 772 . . . . . . 7 ((A x (x = {y} x B)) ↔ (x = {y} (A x x B)))
1716exbii 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 xA {y}))
20 vex 2862 . . . . . . . . . 10 y V
2120elsnc2 3762 . . . . . . . . 9 (A {y} ↔ A = y)
2219, 21syl6bb 252 . . . . . . . 8 (x = {y} → (A xA = y))
23 eleq1 2413 . . . . . . . 8 (x = {y} → (x B ↔ {y} B))
2422, 23anbi12d 691 . . . . . . 7 (x = {y} → ((A x x B) ↔ (A = y {y} B)))
2518, 24ceqsexv 2894 . . . . . 6 (x(x = {y} (A x x B)) ↔ (A = y {y} B))
26 eqcom 2355 . . . . . . 7 (A = yy = A)
2726anbi1i 676 . . . . . 6 ((A = y {y} B) ↔ (y = A {y} B))
2817, 25, 273bitri 262 . . . . 5 (x(A x (x = {y} x B)) ↔ (y = A {y} B))
2928exbii 1582 . . . 4 (yx(A x (x = {y} x B)) ↔ y(y = A {y} B))
3014, 15, 293bitri 262 . . 3 (x(A x x (B ∩ 1c)) ↔ y(y = A {y} B))
312, 3, 303bitri 262 . 2 (A 1By(y = A {y} B))
32 sneq 3744 . . . 4 (y = A → {y} = {A})
3332eleq1d 2419 . . 3 (y = A → ({y} B ↔ {A} B))
3433ceqsexgv 2971 . 2 (A V → (y(y = A {y} B) ↔ {A} B))
3531, 34syl5bb 248 1 (A V → (A 1B ↔ {A} B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  {csn 3737  ∪cuni 3891  ⋃1cuni1 4133  1cc1c 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
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