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

Theorem uneqin 3854
Description: Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
uneqin ((𝐴𝐵) = (𝐴𝐵) ↔ 𝐴 = 𝐵)

Proof of Theorem uneqin
StepHypRef Expression
1 eqimss 3636 . . . 4 ((𝐴𝐵) = (𝐴𝐵) → (𝐴𝐵) ⊆ (𝐴𝐵))
2 unss 3765 . . . . 5 ((𝐴 ⊆ (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) ↔ (𝐴𝐵) ⊆ (𝐴𝐵))
3 ssin 3813 . . . . . . 7 ((𝐴𝐴𝐴𝐵) ↔ 𝐴 ⊆ (𝐴𝐵))
4 sstr 3591 . . . . . . 7 ((𝐴𝐴𝐴𝐵) → 𝐴𝐵)
53, 4sylbir 225 . . . . . 6 (𝐴 ⊆ (𝐴𝐵) → 𝐴𝐵)
6 ssin 3813 . . . . . . 7 ((𝐵𝐴𝐵𝐵) ↔ 𝐵 ⊆ (𝐴𝐵))
7 simpl 473 . . . . . . 7 ((𝐵𝐴𝐵𝐵) → 𝐵𝐴)
86, 7sylbir 225 . . . . . 6 (𝐵 ⊆ (𝐴𝐵) → 𝐵𝐴)
95, 8anim12i 589 . . . . 5 ((𝐴 ⊆ (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) → (𝐴𝐵𝐵𝐴))
102, 9sylbir 225 . . . 4 ((𝐴𝐵) ⊆ (𝐴𝐵) → (𝐴𝐵𝐵𝐴))
111, 10syl 17 . . 3 ((𝐴𝐵) = (𝐴𝐵) → (𝐴𝐵𝐵𝐴))
12 eqss 3598 . . 3 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
1311, 12sylibr 224 . 2 ((𝐴𝐵) = (𝐴𝐵) → 𝐴 = 𝐵)
14 unidm 3734 . . . 4 (𝐴𝐴) = 𝐴
15 inidm 3800 . . . 4 (𝐴𝐴) = 𝐴
1614, 15eqtr4i 2646 . . 3 (𝐴𝐴) = (𝐴𝐴)
17 uneq2 3739 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
18 ineq2 3786 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
1916, 17, 183eqtr3a 2679 . 2 (𝐴 = 𝐵 → (𝐴𝐵) = (𝐴𝐵))
2013, 19impbii 199 1 ((𝐴𝐵) = (𝐴𝐵) ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  cun 3553  cin 3554  wss 3555
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-v 3188  df-un 3560  df-in 3562  df-ss 3569
This theorem is referenced by:  uniintsn  4479
  Copyright terms: Public domain W3C validator