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Theorem infdif2 7836
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 6987 . . . . . . 7  |-  ( ( A  \  B )  ~<_  B  ->  -.  B  ~<  ( A  \  B
) )
2 simp3 957 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  A )
3 infdif 7835 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~~  A )
4 ensym 6910 . . . . . . . . 9  |-  ( ( A  \  B ) 
~~  A  ->  A  ~~  ( A  \  B
) )
53, 4syl 15 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~~  ( A  \  B
) )
6 sdomentr 6995 . . . . . . . 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 7622 . . . 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 3303 . . . 4  |-  ( A 
\  B )  C_  A
17 ssdomg 6907 . . . 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 6914 . . . 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 1684    \ cdif 3149    C_ wss 3152   class class class wbr 4023   omcom 4656   dom cdm 4689    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568
This theorem is referenced by:  axgroth3  8453
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 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-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-cda 7794
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