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Theorem soss 4332
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 4316 . . 3  |-  ( A 
C_  B  ->  ( R  Po  B  ->  R  Po  A ) )
2 ssel 3174 . . . . . . 7  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
3 ssel 3174 . . . . . . 7  |-  ( A 
C_  B  ->  (
y  e.  A  -> 
y  e.  B ) )
42, 3anim12d 546 . . . . . 6  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  A
)  ->  ( x  e.  B  /\  y  e.  B ) ) )
54imim1d 69 . . . . 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 1609 . . . 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 2580 . . . 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 2580 . . . 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 261 . . 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 546 . 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 4315 . 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 4315 . 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 261 1  |-  ( A 
C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023    Po wpo 4312    Or wor 4313
This theorem is referenced by:  soeq2  4334  wess  4380  wereu  4389  wereu2  4390  ordunifi  7107  fisup2g  7217  fisupcl  7218  ordtypelem8  7240  wemapso2  7267  iunfictbso  7741  fin1a2lem10  8035  fin1a2lem11  8036  zornn0g  8132  ltsopi  8512  npomex  8620  fimaxre  9701  isercolllem1  12138  summolem2  12189  zsum  12191  gsumval3  15191  iccpnfhmeo  18443  xrhmeo  18444  dvgt0lem2  19350  dgrval  19610  dgrcl  19615  dgrub  19616  dgrlb  19618  aannenlem3  19710  logccv  20010  ssnnssfz  23277  xrge0iifiso  23317  erdszelem4  23725  erdszelem8  23729  erdsze2lem1  23734  erdsze2lem2  23735  supfz  24094  inffz  24095  rencldnfilem  26903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-in 3159  df-ss 3166  df-po 4314  df-so 4315
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