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Theorem uneqin 3165
 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 2974 . . . 4 ((AB) = (AB) → (AB) ⊆ (AB))
2 unss 3094 . . . . 5 ((A ⊆ (AB) B ⊆ (AB)) ↔ (AB) ⊆ (AB))
3 ssin 3136 . . . . . . 7 ((AA AB) ↔ A ⊆ (AB))
4 sstr 2930 . . . . . . 7 ((AA AB) → AB)
53, 4sylbir 125 . . . . . 6 (A ⊆ (AB) → AB)
6 ssin 3136 . . . . . . 7 ((BA BB) ↔ B ⊆ (AB))
7 simpl 102 . . . . . . 7 ((BA BB) → BA)
86, 7sylbir 125 . . . . . 6 (B ⊆ (AB) → BA)
95, 8anim12i 321 . . . . 5 ((A ⊆ (AB) B ⊆ (AB)) → (AB BA))
102, 9sylbir 125 . . . 4 ((AB) ⊆ (AB) → (AB BA))
111, 10syl 14 . . 3 ((AB) = (AB) → (AB BA))
12 eqss 2937 . . 3 (A = B ↔ (AB BA))
1311, 12sylibr 137 . 2 ((AB) = (AB) → A = B)
14 unidm 3063 . . . 4 (AA) = A
15 inidm 3123 . . . 4 (AA) = A
1614, 15eqtr4i 2045 . . 3 (AA) = (AA)
17 uneq2 3068 . . 3 (A = B → (AA) = (AB))
18 ineq2 3109 . . 3 (A = B → (AA) = (AB))
1916, 17, 183eqtr3a 2078 . 2 (A = B → (AB) = (AB))
2013, 19impbii 117 1 ((AB) = (AB) ↔ A = B)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228   ∪ cun 2892   ∩ cin 2893   ⊆ wss 2894 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908 This theorem is referenced by: (None)
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