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Theorem uneqin 3506
 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 3323 . . . 4 ((AB) = (AB) → (AB) (AB))
2 unss 3437 . . . . 5 ((A (AB) B (AB)) ↔ (AB) (AB))
3 ssin 3477 . . . . . . 7 ((A A A B) ↔ A (AB))
4 sstr 3280 . . . . . . 7 ((A A A B) → A B)
53, 4sylbir 204 . . . . . 6 (A (AB) → A B)
6 ssin 3477 . . . . . . 7 ((B A B B) ↔ B (AB))
7 simpl 443 . . . . . . 7 ((B A B B) → B A)
86, 7sylbir 204 . . . . . 6 (B (AB) → B A)
95, 8anim12i 549 . . . . 5 ((A (AB) B (AB)) → (A B B A))
102, 9sylbir 204 . . . 4 ((AB) (AB) → (A B B A))
111, 10syl 15 . . 3 ((AB) = (AB) → (A B B A))
12 eqss 3287 . . 3 (A = B ↔ (A B B A))
1311, 12sylibr 203 . 2 ((AB) = (AB) → A = B)
14 unidm 3407 . . . 4 (AA) = A
15 inidm 3464 . . . 4 (AA) = A
1614, 15eqtr4i 2376 . . 3 (AA) = (AA)
17 uneq2 3412 . . 3 (A = B → (AA) = (AB))
18 ineq2 3451 . . 3 (A = B → (AA) = (AB))
1916, 17, 183eqtr3a 2409 . 2 (A = B → (AB) = (AB))
2013, 19impbii 180 1 ((AB) = (AB) ↔ A = B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259 This theorem is referenced by:  uniintsn  3963
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