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Theorem ssequn1 3433
 Description: A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssequn1 (A B ↔ (AB) = B)

Proof of Theorem ssequn1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 bicom 191 . . . 4 ((x B ↔ (x A x B)) ↔ ((x A x B) ↔ x B))
2 pm4.72 846 . . . 4 ((x Ax B) ↔ (x B ↔ (x A x B)))
3 elun 3220 . . . . 5 (x (AB) ↔ (x A x B))
43bibi1i 305 . . . 4 ((x (AB) ↔ x B) ↔ ((x A x B) ↔ x B))
51, 2, 43bitr4i 268 . . 3 ((x Ax B) ↔ (x (AB) ↔ x B))
65albii 1566 . 2 (x(x Ax B) ↔ x(x (AB) ↔ x B))
7 dfss2 3262 . 2 (A Bx(x Ax B))
8 dfcleq 2347 . 2 ((AB) = Bx(x (AB) ↔ x B))
96, 7, 83bitr4i 268 1 (A B ↔ (AB) = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357  ∀wal 1540   = wceq 1642   ∈ 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:  ssequn2  3436  undif  3630  unsneqsn  3887  dflec2  6210
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