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Theorem uneqin 3386
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 3209 . . . 4 ((𝐴𝐵) = (𝐴𝐵) → (𝐴𝐵) ⊆ (𝐴𝐵))
2 unss 3309 . . . . 5 ((𝐴 ⊆ (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) ↔ (𝐴𝐵) ⊆ (𝐴𝐵))
3 ssin 3357 . . . . . . 7 ((𝐴𝐴𝐴𝐵) ↔ 𝐴 ⊆ (𝐴𝐵))
4 sstr 3163 . . . . . . 7 ((𝐴𝐴𝐴𝐵) → 𝐴𝐵)
53, 4sylbir 135 . . . . . 6 (𝐴 ⊆ (𝐴𝐵) → 𝐴𝐵)
6 ssin 3357 . . . . . . 7 ((𝐵𝐴𝐵𝐵) ↔ 𝐵 ⊆ (𝐴𝐵))
7 simpl 109 . . . . . . 7 ((𝐵𝐴𝐵𝐵) → 𝐵𝐴)
86, 7sylbir 135 . . . . . 6 (𝐵 ⊆ (𝐴𝐵) → 𝐵𝐴)
95, 8anim12i 338 . . . . 5 ((𝐴 ⊆ (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) → (𝐴𝐵𝐵𝐴))
102, 9sylbir 135 . . . 4 ((𝐴𝐵) ⊆ (𝐴𝐵) → (𝐴𝐵𝐵𝐴))
111, 10syl 14 . . 3 ((𝐴𝐵) = (𝐴𝐵) → (𝐴𝐵𝐵𝐴))
12 eqss 3170 . . 3 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
1311, 12sylibr 134 . 2 ((𝐴𝐵) = (𝐴𝐵) → 𝐴 = 𝐵)
14 unidm 3278 . . . 4 (𝐴𝐴) = 𝐴
15 inidm 3344 . . . 4 (𝐴𝐴) = 𝐴
1614, 15eqtr4i 2201 . . 3 (𝐴𝐴) = (𝐴𝐴)
17 uneq2 3283 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
18 ineq2 3330 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
1916, 17, 183eqtr3a 2234 . 2 (𝐴 = 𝐵 → (𝐴𝐵) = (𝐴𝐵))
2013, 19impbii 126 1 ((𝐴𝐵) = (𝐴𝐵) ↔ 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  cun 3127  cin 3128  wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142
This theorem is referenced by: (None)
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