ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssint GIF version

Theorem ssint 3847
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 3137 . 2 (𝐴 𝐵 ↔ ∀𝑦𝐴 𝑦 𝐵)
2 vex 2733 . . . 4 𝑦 ∈ V
32elint2 3838 . . 3 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43ralbii 2476 . 2 (∀𝑦𝐴 𝑦 𝐵 ↔ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
5 ralcom 2633 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
6 dfss3 3137 . . . 4 (𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
76ralbii 2476 . . 3 (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
85, 7bitr4i 186 . 2 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥)
91, 4, 83bitri 205 1 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 2141  wral 2448  wss 3121   cint 3831
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-int 3832
This theorem is referenced by:  ssintab  3848  ssintub  3849  iinpw  3963  trint  4102  fintm  5383  bj-ssom  13971
  Copyright terms: Public domain W3C validator