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Theorem uneqin 4294
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 4053 . . . 4 ((𝐴𝐵) = (𝐴𝐵) → (𝐴𝐵) ⊆ (𝐴𝐵))
2 unss 4199 . . . . 5 ((𝐴 ⊆ (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) ↔ (𝐴𝐵) ⊆ (𝐴𝐵))
3 ssin 4246 . . . . . . 7 ((𝐴𝐴𝐴𝐵) ↔ 𝐴 ⊆ (𝐴𝐵))
4 sstr 4003 . . . . . . 7 ((𝐴𝐴𝐴𝐵) → 𝐴𝐵)
53, 4sylbir 235 . . . . . 6 (𝐴 ⊆ (𝐴𝐵) → 𝐴𝐵)
6 ssin 4246 . . . . . . 7 ((𝐵𝐴𝐵𝐵) ↔ 𝐵 ⊆ (𝐴𝐵))
7 simpl 482 . . . . . . 7 ((𝐵𝐴𝐵𝐵) → 𝐵𝐴)
86, 7sylbir 235 . . . . . 6 (𝐵 ⊆ (𝐴𝐵) → 𝐵𝐴)
95, 8anim12i 613 . . . . 5 ((𝐴 ⊆ (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) → (𝐴𝐵𝐵𝐴))
102, 9sylbir 235 . . . 4 ((𝐴𝐵) ⊆ (𝐴𝐵) → (𝐴𝐵𝐵𝐴))
111, 10syl 17 . . 3 ((𝐴𝐵) = (𝐴𝐵) → (𝐴𝐵𝐵𝐴))
12 eqss 4010 . . 3 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
1311, 12sylibr 234 . 2 ((𝐴𝐵) = (𝐴𝐵) → 𝐴 = 𝐵)
14 unidm 4166 . . . 4 (𝐴𝐴) = 𝐴
15 inidm 4234 . . . 4 (𝐴𝐴) = 𝐴
1614, 15eqtr4i 2765 . . 3 (𝐴𝐴) = (𝐴𝐴)
17 uneq2 4171 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
18 ineq2 4221 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
1916, 17, 183eqtr3a 2798 . 2 (𝐴 = 𝐵 → (𝐴𝐵) = (𝐴𝐵))
2013, 19impbii 209 1 ((𝐴𝐵) = (𝐴𝐵) ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  cun 3960  cin 3961  wss 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-un 3967  df-in 3969  df-ss 3979
This theorem is referenced by:  uniintsn  4989
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