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Theorem intss 3636
 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 70 . . . . 5 ((𝑦𝐴𝑦𝐵) → ((𝑦𝐵𝑥𝑦) → (𝑦𝐴𝑥𝑦)))
21al2imi 1347 . . . 4 (∀𝑦(𝑦𝐴𝑦𝐵) → (∀𝑦(𝑦𝐵𝑥𝑦) → ∀𝑦(𝑦𝐴𝑥𝑦)))
3 vex 2560 . . . . 5 𝑥 ∈ V
43elint 3621 . . . 4 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
53elint 3621 . . . 4 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
62, 4, 53imtr4g 194 . . 3 (∀𝑦(𝑦𝐴𝑦𝐵) → (𝑥 𝐵𝑥 𝐴))
76alrimiv 1754 . 2 (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑥(𝑥 𝐵𝑥 𝐴))
8 dfss2 2934 . 2 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 2934 . 2 ( 𝐵 𝐴 ↔ ∀𝑥(𝑥 𝐵𝑥 𝐴))
107, 8, 93imtr4i 190 1 (𝐴𝐵 𝐵 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241   ∈ wcel 1393   ⊆ wss 2917  ∩ cint 3615 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-int 3616 This theorem is referenced by: (None)
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