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Theorem ssunieq 3924
 Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.)
Assertion
Ref Expression
ssunieq ((A B x B x A) → A = B)
Distinct variable groups:   x,A   x,B

Proof of Theorem ssunieq
StepHypRef Expression
1 elssuni 3919 . . 3 (A BA B)
2 unissb 3921 . . . 4 (B Ax B x A)
32biimpri 197 . . 3 (x B x AB A)
41, 3anim12i 549 . 2 ((A B x B x A) → (A B B A))
5 eqss 3287 . 2 (A = B ↔ (A B B A))
64, 5sylibr 203 1 ((A B x B x A) → A = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ 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:  unimax  3925
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