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Theorem intss 3880
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 1469 . . . 4 (∀𝑦(𝑦𝐴𝑦𝐵) → (∀𝑦(𝑦𝐵𝑥𝑦) → ∀𝑦(𝑦𝐴𝑥𝑦)))
3 vex 2755 . . . . 5 𝑥 ∈ V
43elint 3865 . . . 4 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
53elint 3865 . . . 4 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
62, 4, 53imtr4g 205 . . 3 (∀𝑦(𝑦𝐴𝑦𝐵) → (𝑥 𝐵𝑥 𝐴))
76alrimiv 1885 . 2 (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑥(𝑥 𝐵𝑥 𝐴))
8 dfss2 3159 . 2 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 3159 . 2 ( 𝐵 𝐴 ↔ ∀𝑥(𝑥 𝐵𝑥 𝐴))
107, 8, 93imtr4i 201 1 (𝐴𝐵 𝐵 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  wcel 2160  wss 3144   cint 3859
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-int 3860
This theorem is referenced by:  lspss  13715  clsss  14078
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