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Theorem intss 3762
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 1419 . . . 4 (∀𝑦(𝑦𝐴𝑦𝐵) → (∀𝑦(𝑦𝐵𝑥𝑦) → ∀𝑦(𝑦𝐴𝑥𝑦)))
3 vex 2663 . . . . 5 𝑥 ∈ V
43elint 3747 . . . 4 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
53elint 3747 . . . 4 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
62, 4, 53imtr4g 204 . . 3 (∀𝑦(𝑦𝐴𝑦𝐵) → (𝑥 𝐵𝑥 𝐴))
76alrimiv 1830 . 2 (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑥(𝑥 𝐵𝑥 𝐴))
8 dfss2 3056 . 2 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 3056 . 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 1314  wcel 1465  wss 3041   cint 3741
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-int 3742
This theorem is referenced by:  clsss  12214
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