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Theorem intss 3787
 Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
intss (𝐴𝐵 𝐵 𝐴)

Proof of Theorem intss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imim1 76 . . . . 5 ((𝑦𝐴𝑦𝐵) → ((𝑦𝐵𝑥𝑦) → (𝑦𝐴𝑥𝑦)))
21al2imi 1434 . . . 4 (∀𝑦(𝑦𝐴𝑦𝐵) → (∀𝑦(𝑦𝐵𝑥𝑦) → ∀𝑦(𝑦𝐴𝑥𝑦)))
3 vex 2684 . . . . 5 𝑥 ∈ V
43elint 3772 . . . 4 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
53elint 3772 . . . 4 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
62, 4, 53imtr4g 204 . . 3 (∀𝑦(𝑦𝐴𝑦𝐵) → (𝑥 𝐵𝑥 𝐴))
76alrimiv 1846 . 2 (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑥(𝑥 𝐵𝑥 𝐴))
8 dfss2 3081 . 2 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 3081 . 2 ( 𝐵 𝐴 ↔ ∀𝑥(𝑥 𝐵𝑥 𝐴))
107, 8, 93imtr4i 200 1 (𝐴𝐵 𝐵 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1329   ∈ wcel 1480   ⊆ wss 3066  ∩ cint 3766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-int 3767 This theorem is referenced by:  clsss  12276
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