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Theorem unissb 3580
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 ( ABx A xB)
Distinct variable groups:   x,A   x,B

Proof of Theorem unissb
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eluni 3553 . . . . . 6 (y Ax(y x x A))
21imbi1i 227 . . . . 5 ((y Ay B) ↔ (x(y x x A) → y B))
3 19.23v 1741 . . . . 5 (x((y x x A) → y B) ↔ (x(y x x A) → y B))
42, 3bitr4i 176 . . . 4 ((y Ay B) ↔ x((y x x A) → y B))
54albii 1335 . . 3 (y(y Ay B) ↔ yx((y x x A) → y B))
6 alcom 1343 . . . 4 (yx((y x x A) → y B) ↔ xy((y x x A) → y B))
7 19.21v 1731 . . . . . 6 (y(x A → (y xy B)) ↔ (x Ay(y xy B)))
8 impexp 250 . . . . . . . 8 (((y x x A) → y B) ↔ (y x → (x Ay B)))
9 bi2.04 237 . . . . . . . 8 ((y x → (x Ay B)) ↔ (x A → (y xy B)))
108, 9bitri 173 . . . . . . 7 (((y x x A) → y B) ↔ (x A → (y xy B)))
1110albii 1335 . . . . . 6 (y((y x x A) → y B) ↔ y(x A → (y xy B)))
12 dfss2 2907 . . . . . . 7 (xBy(y xy B))
1312imbi2i 215 . . . . . 6 ((x AxB) ↔ (x Ay(y xy B)))
147, 11, 133bitr4i 201 . . . . 5 (y((y x x A) → y B) ↔ (x AxB))
1514albii 1335 . . . 4 (xy((y x x A) → y B) ↔ x(x AxB))
166, 15bitri 173 . . 3 (yx((y x x A) → y B) ↔ x(x AxB))
175, 16bitri 173 . 2 (y(y Ay B) ↔ x(x AxB))
18 dfss2 2907 . 2 ( ABy(y Ay B))
19 df-ral 2285 . 2 (x A xBx(x AxB))
2017, 18, 193bitr4i 201 1 ( ABx A xB)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224  wex 1358   wcel 1370  wral 2280  wss 2890   cuni 3550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-in 2897  df-ss 2904  df-uni 3551
This theorem is referenced by:  uniss2  3581  ssunieq  3583  sspwuni  3709  pwssb  3710  bm2.5ii  4168
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