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Theorem infdif2 8046
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 7192 . . . . . . 7  |-  ( ( A  \  B )  ~<_  B  ->  -.  B  ~<  ( A  \  B
) )
2 simp3 959 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  A )
3 infdif 8045 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~~  A )
43ensymd 7117 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~~  ( A  \  B
) )
5 sdomentr 7200 . . . . . . . 8  |-  ( ( B  ~<  A  /\  A  ~~  ( A  \  B ) )  ->  B  ~<  ( A  \  B ) )
62, 4, 5syl2anc 643 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  ( A  \  B
) )
71, 6nsyl3 113 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  -.  ( A  \  B )  ~<_  B )
873expia 1155 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( B  ~<  A  ->  -.  ( A  \  B
)  ~<_  B ) )
983adant2 976 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( B  ~<  A  ->  -.  ( A  \  B
)  ~<_  B ) )
109con2d 109 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  \  B )  ~<_  B  ->  -.  B  ~<  A ) )
11 domtri2 7832 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
12113adant3 977 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  ~<_  B  <->  -.  B  ~<  A ) )
1310, 12sylibrd 226 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  \  B )  ~<_  B  ->  A  ~<_  B ) )
14 simp1 957 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  A  e.  dom  card )
15 difss 3434 . . . 4  |-  ( A 
\  B )  C_  A
16 ssdomg 7112 . . . 4  |-  ( A  e.  dom  card  ->  ( ( A  \  B
)  C_  A  ->  ( A  \  B )  ~<_  A ) )
1714, 15, 16ee10 1382 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  \  B
)  ~<_  A )
18 domtr 7119 . . . 4  |-  ( ( ( A  \  B
)  ~<_  A  /\  A  ~<_  B )  ->  ( A  \  B )  ~<_  B )
1918ex 424 . . 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 184 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 177    /\ w3a 936    e. wcel 1721    \ cdif 3277    C_ wss 3280   class class class wbr 4172   omcom 4804   dom cdm 4837    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067   cardccrd 7778
This theorem is referenced by:  axgroth3  8662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-cda 8004
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