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Theorem soss 4523
Description: Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
soss  |-  ( A 
C_  B  ->  ( R  Or  B  ->  R  Or  A ) )

Proof of Theorem soss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poss 4507 . . 3  |-  ( A 
C_  B  ->  ( R  Po  B  ->  R  Po  A ) )
2 ssel 3344 . . . . . . 7  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
3 ssel 3344 . . . . . . 7  |-  ( A 
C_  B  ->  (
y  e.  A  -> 
y  e.  B ) )
42, 3anim12d 548 . . . . . 6  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  A
)  ->  ( x  e.  B  /\  y  e.  B ) ) )
54imim1d 72 . . . . 5  |-  ( A 
C_  B  ->  (
( ( x  e.  B  /\  y  e.  B )  ->  (
x R y  \/  x  =  y  \/  y R x ) )  ->  ( (
x  e.  A  /\  y  e.  A )  ->  ( x R y  \/  x  =  y  \/  y R x ) ) ) )
652alimdv 1634 . . . 4  |-  ( A 
C_  B  ->  ( A. x A. y ( ( x  e.  B  /\  y  e.  B
)  ->  ( x R y  \/  x  =  y  \/  y R x ) )  ->  A. x A. y
( ( x  e.  A  /\  y  e.  A )  ->  (
x R y  \/  x  =  y  \/  y R x ) ) ) )
7 r2al 2744 . . . 4  |-  ( A. x  e.  B  A. y  e.  B  (
x R y  \/  x  =  y  \/  y R x )  <->  A. x A. y ( ( x  e.  B  /\  y  e.  B
)  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
8 r2al 2744 . . . 4  |-  ( A. x  e.  A  A. y  e.  A  (
x R y  \/  x  =  y  \/  y R x )  <->  A. x A. y ( ( x  e.  A  /\  y  e.  A
)  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
96, 7, 83imtr4g 263 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  B  A. y  e.  B  ( x R y  \/  x  =  y  \/  y R x )  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) ) )
101, 9anim12d 548 . 2  |-  ( A 
C_  B  ->  (
( R  Po  B  /\  A. x  e.  B  A. y  e.  B  ( x R y  \/  x  =  y  \/  y R x ) )  ->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  (
x R y  \/  x  =  y  \/  y R x ) ) ) )
11 df-so 4506 . 2  |-  ( R  Or  B  <->  ( R  Po  B  /\  A. x  e.  B  A. y  e.  B  ( x R y  \/  x  =  y  \/  y R x ) ) )
12 df-so 4506 . 2  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) ) )
1310, 11, 123imtr4g 263 1  |-  ( A 
C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    \/ w3o 936   A.wal 1550    e. wcel 1726   A.wral 2707    C_ wss 3322   class class class wbr 4214    Po wpo 4503    Or wor 4504
This theorem is referenced by:  soeq2  4525  wess  4571  wereu  4580  wereu2  4581  ordunifi  7359  fisup2g  7473  fisupcl  7474  ordtypelem8  7496  wemapso2  7523  iunfictbso  7997  fin1a2lem10  8291  fin1a2lem11  8292  zornn0g  8387  ltsopi  8767  npomex  8875  fimaxre  9957  isercolllem1  12460  summolem2  12512  zsum  12514  gsumval3  15516  iccpnfhmeo  18972  xrhmeo  18973  dvgt0lem2  19889  dgrval  20149  dgrcl  20154  dgrub  20155  dgrlb  20157  aannenlem3  20249  logccv  20556  ssnnssfz  24150  xrge0iifiso  24323  erdszelem4  24882  erdszelem8  24886  erdsze2lem1  24891  erdsze2lem2  24892  supfz  25201  inffz  25202  prodmolem2  25263  zprod  25265  rencldnfilem  26883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-in 3329  df-ss 3336  df-po 4505  df-so 4506
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