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Theorem infdif2 8091
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 7234 . . . . . . 7  |-  ( ( A  \  B )  ~<_  B  ->  -.  B  ~<  ( A  \  B
) )
2 simp3 960 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  A )
3 infdif 8090 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~~  A )
43ensymd 7159 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~~  ( A  \  B
) )
5 sdomentr 7242 . . . . . . . 8  |-  ( ( B  ~<  A  /\  A  ~~  ( A  \  B ) )  ->  B  ~<  ( A  \  B ) )
62, 4, 5syl2anc 644 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  ( A  \  B
) )
71, 6nsyl3 114 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  -.  ( A  \  B )  ~<_  B )
873expia 1156 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( B  ~<  A  ->  -.  ( A  \  B
)  ~<_  B ) )
983adant2 977 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( B  ~<  A  ->  -.  ( A  \  B
)  ~<_  B ) )
109con2d 110 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  \  B )  ~<_  B  ->  -.  B  ~<  A ) )
11 domtri2 7877 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
12113adant3 978 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  ~<_  B  <->  -.  B  ~<  A ) )
1310, 12sylibrd 227 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  \  B )  ~<_  B  ->  A  ~<_  B ) )
14 simp1 958 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  A  e.  dom  card )
15 difss 3475 . . . 4  |-  ( A 
\  B )  C_  A
16 ssdomg 7154 . . . 4  |-  ( A  e.  dom  card  ->  ( ( A  \  B
)  C_  A  ->  ( A  \  B )  ~<_  A ) )
1714, 15, 16ee10 1386 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  \  B
)  ~<_  A )
18 domtr 7161 . . . 4  |-  ( ( ( A  \  B
)  ~<_  A  /\  A  ~<_  B )  ->  ( A  \  B )  ~<_  B )
1918ex 425 . . 3  |-  ( ( A  \  B )  ~<_  A  ->  ( A  ~<_  B  ->  ( A  \  B )  ~<_  B ) )
2017, 19syl 16 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  ~<_  B  -> 
( A  \  B
)  ~<_  B ) )
2113, 20impbid 185 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 178    /\ w3a 937    e. wcel 1726    \ cdif 3318    C_ wss 3321   class class class wbr 4213   omcom 4846   dom cdm 4879    ~~ cen 7107    ~<_ cdom 7108    ~< csdm 7109   cardccrd 7823
This theorem is referenced by:  axgroth3  8707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-oi 7480  df-card 7827  df-cda 8049
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