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Theorem ssunieq 4838
 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 4833 . . 3 (𝐴𝐵𝐴 𝐵)
2 unissb 4835 . . . 4 ( 𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
32biimpri 231 . . 3 (∀𝑥𝐵 𝑥𝐴 𝐵𝐴)
41, 3anim12i 615 . 2 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → (𝐴 𝐵 𝐵𝐴))
5 eqss 3933 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
64, 5sylibr 237 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109   ⊆ wss 3884  ∪ cuni 4803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-in 3891  df-ss 3901  df-uni 4804 This theorem is referenced by:  unimax  4839  shsspwh  29033
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