Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssunieq Structured version   Visualization version   GIF version

Theorem ssunieq 4438
 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 4433 . . 3 (𝐴𝐵𝐴 𝐵)
2 unissb 4435 . . . 4 ( 𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
32biimpri 218 . . 3 (∀𝑥𝐵 𝑥𝐴 𝐵𝐴)
41, 3anim12i 589 . 2 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → (𝐴 𝐵 𝐵𝐴))
5 eqss 3598 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
64, 5sylibr 224 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907   ⊆ wss 3555  ∪ cuni 4402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-in 3562  df-ss 3569  df-uni 4403 This theorem is referenced by:  unimax  4439  shsspwh  27952
 Copyright terms: Public domain W3C validator