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Theorem sndw 25100
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 4160 . . . . 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 6907 . . . 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 6907 . . . . 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 7004 . . . . 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 6981 . . 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 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  sndw2  25101
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-er 6660  df-en 6864  df-dom 6865
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