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Theorem unissb 3921
 Description: Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
unissb (A Bx A x B)
Distinct variable groups:   x,A   x,B

Proof of Theorem unissb
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eluni 3894 . . . . . 6 (y Ax(y x x A))
21imbi1i 315 . . . . 5 ((y Ay B) ↔ (x(y x x A) → y B))
3 19.23v 1891 . . . . 5 (x((y x x A) → y B) ↔ (x(y x x A) → y B))
42, 3bitr4i 243 . . . 4 ((y Ay B) ↔ x((y x x A) → y B))
54albii 1566 . . 3 (y(y Ay B) ↔ yx((y x x A) → y B))
6 alcom 1737 . . . 4 (yx((y x x A) → y B) ↔ xy((y x x A) → y B))
7 19.21v 1890 . . . . . 6 (y(x A → (y xy B)) ↔ (x Ay(y xy B)))
8 impexp 433 . . . . . . . 8 (((y x x A) → y B) ↔ (y x → (x Ay B)))
9 bi2.04 350 . . . . . . . 8 ((y x → (x Ay B)) ↔ (x A → (y xy B)))
108, 9bitri 240 . . . . . . 7 (((y x x A) → y B) ↔ (x A → (y xy B)))
1110albii 1566 . . . . . 6 (y((y x x A) → y B) ↔ y(x A → (y xy B)))
12 dfss2 3262 . . . . . . 7 (x By(y xy B))
1312imbi2i 303 . . . . . 6 ((x Ax B) ↔ (x Ay(y xy B)))
147, 11, 133bitr4i 268 . . . . 5 (y((y x x A) → y B) ↔ (x Ax B))
1514albii 1566 . . . 4 (xy((y x x A) → y B) ↔ x(x Ax B))
166, 15bitri 240 . . 3 (yx((y x x A) → y B) ↔ x(x Ax B))
175, 16bitri 240 . 2 (y(y Ay B) ↔ x(x Ax B))
18 dfss2 3262 . 2 (A By(y Ay B))
19 df-ral 2619 . 2 (x A x Bx(x Ax B))
2017, 18, 193bitr4i 268 1 (A Bx A x B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257  ∪cuni 3891 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 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-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-uni 3892 This theorem is referenced by:  uniss2  3922  ssunieq  3924  sspwuni  4051  pwssb  4052
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