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Theorem infdif2 7852
Description: Cardinality ordering for an infinite set difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infdif2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  \  B )  ~<_  B  <->  A  ~<_  B ) )

Proof of Theorem infdif2
StepHypRef Expression
1 domnsym 7003 . . . . . . 7  |-  ( ( A  \  B )  ~<_  B  ->  -.  B  ~<  ( A  \  B
) )
2 simp3 957 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  A )
3 infdif 7851 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~~  A )
4 ensym 6926 . . . . . . . . 9  |-  ( ( A  \  B ) 
~~  A  ->  A  ~~  ( A  \  B
) )
53, 4syl 15 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~~  ( A  \  B
) )
6 sdomentr 7011 . . . . . . . 8  |-  ( ( B  ~<  A  /\  A  ~~  ( A  \  B ) )  ->  B  ~<  ( A  \  B ) )
72, 5, 6syl2anc 642 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  ( A  \  B
) )
81, 7nsyl3 111 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  -.  ( A  \  B )  ~<_  B )
983expia 1153 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( B  ~<  A  ->  -.  ( A  \  B
)  ~<_  B ) )
1093adant2 974 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( B  ~<  A  ->  -.  ( A  \  B
)  ~<_  B ) )
1110con2d 107 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  \  B )  ~<_  B  ->  -.  B  ~<  A ) )
12 domtri2 7638 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
13123adant3 975 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  ~<_  B  <->  -.  B  ~<  A ) )
1411, 13sylibrd 225 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  \  B )  ~<_  B  ->  A  ~<_  B ) )
15 simp1 955 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  A  e.  dom  card )
16 difss 3316 . . . 4  |-  ( A 
\  B )  C_  A
17 ssdomg 6923 . . . 4  |-  ( A  e.  dom  card  ->  ( ( A  \  B
)  C_  A  ->  ( A  \  B )  ~<_  A ) )
1815, 16, 17ee10 1366 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  \  B
)  ~<_  A )
19 domtr 6930 . . . 4  |-  ( ( ( A  \  B
)  ~<_  A  /\  A  ~<_  B )  ->  ( A  \  B )  ~<_  B )
2019ex 423 . . 3  |-  ( ( A  \  B )  ~<_  A  ->  ( A  ~<_  B  ->  ( A  \  B )  ~<_  B ) )
2118, 20syl 15 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  ~<_  B  -> 
( A  \  B
)  ~<_  B ) )
2214, 21impbid 183 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  \  B )  ~<_  B  <->  A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1696    \ cdif 3162    C_ wss 3165   class class class wbr 4039   omcom 4672   dom cdm 4705    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584
This theorem is referenced by:  axgroth3  8469
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-cda 7810
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