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Theorem uneqin 3182
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 2991 . . . 4 ((AB) = (AB) → (AB) ⊆ (AB))
2 unss 3111 . . . . 5 ((A ⊆ (AB) B ⊆ (AB)) ↔ (AB) ⊆ (AB))
3 ssin 3153 . . . . . . 7 ((AA AB) ↔ A ⊆ (AB))
4 sstr 2947 . . . . . . 7 ((AA AB) → AB)
53, 4sylbir 125 . . . . . 6 (A ⊆ (AB) → AB)
6 ssin 3153 . . . . . . 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 2954 . . 3 (A = B ↔ (AB BA))
1311, 12sylibr 137 . 2 ((AB) = (AB) → A = B)
14 unidm 3080 . . . 4 (AA) = A
15 inidm 3140 . . . 4 (AA) = A
1614, 15eqtr4i 2060 . . 3 (AA) = (AA)
17 uneq2 3085 . . 3 (A = B → (AA) = (AB))
18 ineq2 3126 . . 3 (A = B → (AA) = (AB))
1916, 17, 183eqtr3a 2093 . 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 1242  cun 2909  cin 2910  wss 2911
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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