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Theorem ssint 3689
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 3004 . 2 (𝐴 𝐵 ↔ ∀𝑦𝐴 𝑦 𝐵)
2 vex 2618 . . . 4 𝑦 ∈ V
32elint2 3680 . . 3 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43ralbii 2380 . 2 (∀𝑦𝐴 𝑦 𝐵 ↔ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
5 ralcom 2526 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
6 dfss3 3004 . . . 4 (𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
76ralbii 2380 . . 3 (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
85, 7bitr4i 185 . 2 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥)
91, 4, 83bitri 204 1 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wb 103  wcel 1436  wral 2355  wss 2988   cint 3673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-in 2994  df-ss 3001  df-int 3674
This theorem is referenced by:  ssintab  3690  ssintub  3691  iinpw  3800  trint  3928  fintm  5161  bj-ssom  11300
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