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Theorem ssint 4428
 Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
ssint (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssint
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfss3 3558 . 2 (𝐴 𝐵 ↔ ∀𝑦𝐴 𝑦 𝐵)
2 vex 3176 . . . 4 𝑦 ∈ V
32elint2 4417 . . 3 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43ralbii 2963 . 2 (∀𝑦𝐴 𝑦 𝐵 ↔ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
5 ralcom 3079 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
6 dfss3 3558 . . . 4 (𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
76ralbii 2963 . . 3 (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
85, 7bitr4i 266 . 2 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥)
91, 4, 83bitri 285 1 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547  df-ss 3554  df-int 4411 This theorem is referenced by:  ssintab  4429  ssintub  4430  iinpw  4550  trint  4696  oneqmini  5693  fint  5997  sorpssint  6845  iscard2  8685  coftr  8978  isf32lem2  9059  inttsk  9475  dfrtrcl2  13650  isacs1i  16141  mrelatglb  17007  fbfinnfr  21455  fclscmp  21644  fneint  31513  topmeet  31529  igenval2  33035  ismrcd1  36279  dftrcl3  37031  dfrtrcl3  37044  sssalgen  39229  issalgend  39232
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