Theorem ssrin 2234

Description: Add right intersection to subclass relation.

Assertion

Ref	Expression
ssrin	|- (A (_ B -> (A i^i C) (_ (B i^i C))

Proof of Theorem ssrin

Step	Hyp	Ref	Expression
1		pm3.45 562	. . . 4 |- ((x e. A -> x e. B) -> ((x e. A /\ x e. C) -> (x e. B /\ x e. C)))
2		elin 2207	. . . 4 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
3		elin 2207	. . . 4 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
4	1, 2, 3	3imtr4g 553	. . 3 |- ((x e. A -> x e. B) -> (x e. (A i^i C) -> x e. (B i^i C)))
5	4	19.20i 992	. 2 |- (A.x(x e. A -> x e. B) -> A.x(x e. (A i^i C) -> x e. (B i^i C)))
6		dfss2 2058	. 2 |- (A (_ B <-> A.x(x e. A -> x e. B))
7		dfss2 2058	. 2 |- ((A i^i C) (_ (B i^i C) <-> A.x(x e. (A i^i C) -> x e. (B i^i C)))
8	5, 6, 7	3imtr4 219	1 |- (A (_ B -> (A i^i C) (_ (B i^i C))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958   i^i cin 2046   (_ wss 2047

This theorem is referenced by:  sslin 2235  ss2in 2236  ssdisj 2318  ssres 3385  sbthlem7 4453  phplem2 4509  tgsst 7636  islp2 7747  orthin 9370  3oalem6 9612  mdbr2 10223  mdslle1 10244  mdslle2 10245  mdslj1 10246  mdslj2 10247  mdsl2 10249  mdslmd1lem1 10252  mdslmd1lem2 10253  mdslmd3 10259  mdexch 10262  sumdmdlem 10345

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053

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