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Theorem unissb 3601
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 3574 . . . . . 6 (y Ax(y x x A))
21imbi1i 227 . . . . 5 ((y Ay B) ↔ (x(y x x A) → y B))
3 19.23v 1760 . . . . 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 1356 . . 3 (y(y Ay B) ↔ yx((y x x A) → y B))
6 alcom 1364 . . . 4 (yx((y x x A) → y B) ↔ xy((y x x A) → y B))
7 19.21v 1750 . . . . . 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 1356 . . . . . 6 (y((y x x A) → y B) ↔ y(x A → (y xy B)))
12 dfss2 2928 . . . . . . 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 1356 . . . 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 2928 . 2 ( ABy(y Ay B))
19 df-ral 2305 . 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 1240  wex 1378   wcel 1390  wral 2300  wss 2911   cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572
This theorem is referenced by:  uniss2  3602  ssunieq  3604  sspwuni  3730  pwssb  3731  bm2.5ii  4188
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