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Theorem sndw 25203
Description: If  A is a part of  B and 
B a part of  C and  A is equipotent to  C then  A is equipotent to  B. The art of sandwich applied to set theory. (Contributed by FL, 16-Apr-2011.) (Revised by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
sndw  |-  ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  ->  ( A  ~~  C  ->  A  ~~  B ) )

Proof of Theorem sndw
StepHypRef Expression
1 simpl2 959 . . . . 5  |-  ( ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  /\  A  ~~  C )  ->  B  C_  C
)
2 simpl3 960 . . . . 5  |-  ( ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  /\  A  ~~  C )  ->  C  e.  _V )
3 ssexg 4176 . . . . 5  |-  ( ( B  C_  C  /\  C  e.  _V )  ->  B  e.  _V )
41, 2, 3syl2anc 642 . . . 4  |-  ( ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  /\  A  ~~  C )  ->  B  e.  _V )
5 simpl1 958 . . . 4  |-  ( ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  /\  A  ~~  C )  ->  A  C_  B
)
6 ssdomg 6923 . . . 4  |-  ( B  e.  _V  ->  ( A  C_  B  ->  A  ~<_  B ) )
74, 5, 6sylc 56 . . 3  |-  ( ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  /\  A  ~~  C )  ->  A  ~<_  B )
8 ssdomg 6923 . . . . 5  |-  ( C  e.  _V  ->  ( B  C_  C  ->  B  ~<_  C ) )
92, 1, 8sylc 56 . . . 4  |-  ( ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  /\  A  ~~  C )  ->  B  ~<_  C )
10 domen2 7020 . . . . 5  |-  ( A 
~~  C  ->  ( B  ~<_  A  <->  B  ~<_  C ) )
1110adantl 452 . . . 4  |-  ( ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  /\  A  ~~  C )  ->  ( B  ~<_  A  <-> 
B  ~<_  C ) )
129, 11mpbird 223 . . 3  |-  ( ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  /\  A  ~~  C )  ->  B  ~<_  A )
13 sbth 6997 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
147, 12, 13syl2anc 642 . 2  |-  ( ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  /\  A  ~~  C )  ->  A  ~~  B
)
1514ex 423 1  |-  ( ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  ->  ( A  ~~  C  ->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    ~~ cen 6876    ~<_ cdom 6877
This theorem is referenced by:  sndw2  25204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-er 6676  df-en 6880  df-dom 6881
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