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Theorem unss 3437
Description: The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
Assertion
Ref Expression
unss ((A C B C) ↔ (AB) C)

Proof of Theorem unss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3262 . 2 ((AB) Cx(x (AB) → x C))
2 19.26 1593 . . 3 (x((x Ax C) (x Bx C)) ↔ (x(x Ax C) x(x Bx C)))
3 elun 3220 . . . . . 6 (x (AB) ↔ (x A x B))
43imbi1i 315 . . . . 5 ((x (AB) → x C) ↔ ((x A x B) → x C))
5 jaob 758 . . . . 5 (((x A x B) → x C) ↔ ((x Ax C) (x Bx C)))
64, 5bitri 240 . . . 4 ((x (AB) → x C) ↔ ((x Ax C) (x Bx C)))
76albii 1566 . . 3 (x(x (AB) → x C) ↔ x((x Ax C) (x Bx C)))
8 dfss2 3262 . . . 4 (A Cx(x Ax C))
9 dfss2 3262 . . . 4 (B Cx(x Bx C))
108, 9anbi12i 678 . . 3 ((A C B C) ↔ (x(x Ax C) x(x Bx C)))
112, 7, 103bitr4i 268 . 2 (x(x (AB) → x C) ↔ (A C B C))
121, 11bitr2i 241 1 ((A C B C) ↔ (AB) C)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wo 357   wa 358  wal 1540   wcel 1710  cun 3207   wss 3257
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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259
This theorem is referenced by:  unssi  3438  unssd  3439  unssad  3440  unssbd  3441  nsspssun  3488  uneqin  3506  uneqdifeq  3638  prss  3861  prssg  3862  ssunsn2  3865  tpss  3871  evenoddnnnul  4514  nnadjoinpw  4521  tfinnn  4534  vfinspeqtncv  4553  sbthlem1  6203  spacssnc  6284
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