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Theorem ssint 3858
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 3145 . 2 (𝐴 𝐵 ↔ ∀𝑦𝐴 𝑦 𝐵)
2 vex 2740 . . . 4 𝑦 ∈ V
32elint2 3849 . . 3 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43ralbii 2483 . 2 (∀𝑦𝐴 𝑦 𝐵 ↔ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
5 ralcom 2640 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
6 dfss3 3145 . . . 4 (𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
76ralbii 2483 . . 3 (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
85, 7bitr4i 187 . 2 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥)
91, 4, 83bitri 206 1 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2148  wral 2455  wss 3129   cint 3842
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-int 3843
This theorem is referenced by:  ssintab  3859  ssintub  3860  iinpw  3974  trint  4113  fintm  5397  bj-ssom  14337
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