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Theorem ssunieq 4942
Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.)
Assertion
Ref Expression
ssunieq ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssunieq
StepHypRef Expression
1 elssuni 4936 . . 3 (𝐴𝐵𝐴 𝐵)
2 unissb 4938 . . . 4 ( 𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
32biimpri 228 . . 3 (∀𝑥𝐵 𝑥𝐴 𝐵𝐴)
41, 3anim12i 613 . 2 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → (𝐴 𝐵 𝐵𝐴))
5 eqss 3998 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
64, 5sylibr 234 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  wss 3950   cuni 4906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-v 3481  df-ss 3967  df-uni 4907
This theorem is referenced by:  unimax  4943  shsspwh  31266
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