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Theorem infdif 7851
Description: The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infdif  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~~  A )

Proof of Theorem infdif
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  e.  dom  card )
2 difss 3316 . . 3  |-  ( A 
\  B )  C_  A
3 ssdomg 6923 . . 3  |-  ( A  e.  dom  card  ->  ( ( A  \  B
)  C_  A  ->  ( A  \  B )  ~<_  A ) )
41, 2, 3ee10 1366 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~<_  A )
5 sdomdom 6905 . . . . . . . . 9  |-  ( B 
~<  A  ->  B  ~<_  A )
653ad2ant3 978 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<_  A )
7 numdom 7681 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\  B  ~<_  A )  ->  B  e.  dom  card )
81, 6, 7syl2anc 642 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  e.  dom  card )
9 unnum 7842 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  u.  B )  e.  dom  card )
101, 8, 9syl2anc 642 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  u.  B )  e.  dom  card )
11 ssun1 3351 . . . . . 6  |-  A  C_  ( A  u.  B
)
12 ssdomg 6923 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( A  C_  ( A  u.  B )  ->  A  ~<_  ( A  u.  B
) ) )
1310, 11, 12ee10 1366 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( A  u.  B
) )
14 undif1 3542 . . . . . 6  |-  ( ( A  \  B )  u.  B )  =  ( A  u.  B
)
15 ssnum 7682 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\  ( A  \  B
)  C_  A )  ->  ( A  \  B
)  e.  dom  card )
161, 2, 15sylancl 643 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  e. 
dom  card )
17 uncdadom 7813 . . . . . . 7  |-  ( ( ( A  \  B
)  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( A 
\  B )  u.  B )  ~<_  ( ( A  \  B )  +c  B ) )
1816, 8, 17syl2anc 642 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  u.  B )  ~<_  ( ( A  \  B )  +c  B
) )
1914, 18syl5eqbrr 4073 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  u.  B )  ~<_  ( ( A  \  B )  +c  B
) )
20 domtr 6930 . . . . 5  |-  ( ( A  ~<_  ( A  u.  B )  /\  ( A  u.  B )  ~<_  ( ( A  \  B )  +c  B
) )  ->  A  ~<_  ( ( A  \  B )  +c  B
) )
2113, 19, 20syl2anc 642 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( ( A  \  B )  +c  B
) )
22 simp3 957 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  A )
23 sdomdom 6905 . . . . . . . . 9  |-  ( ( A  \  B ) 
~<  B  ->  ( A 
\  B )  ~<_  B )
24 cdadom1 7828 . . . . . . . . 9  |-  ( ( A  \  B )  ~<_  B  ->  ( ( A  \  B )  +c  B )  ~<_  ( B  +c  B ) )
2523, 24syl 15 . . . . . . . 8  |-  ( ( A  \  B ) 
~<  B  ->  ( ( A  \  B )  +c  B )  ~<_  ( B  +c  B ) )
26 domtr 6930 . . . . . . . . . . 11  |-  ( ( A  ~<_  ( ( A 
\  B )  +c  B )  /\  (
( A  \  B
)  +c  B )  ~<_  ( B  +c  B
) )  ->  A  ~<_  ( B  +c  B
) )
2726ex 423 . . . . . . . . . 10  |-  ( A  ~<_  ( ( A  \  B )  +c  B
)  ->  ( (
( A  \  B
)  +c  B )  ~<_  ( B  +c  B
)  ->  A  ~<_  ( B  +c  B ) ) )
2821, 27syl 15 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( ( A  \  B )  +c  B
)  ~<_  ( B  +c  B )  ->  A  ~<_  ( B  +c  B
) ) )
29 simp2 956 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  om  ~<_  A )
30 domtr 6930 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  ~<_  ( B  +c  B
) )  ->  om  ~<_  ( B  +c  B ) )
3130ex 423 . . . . . . . . . . . 12  |-  ( om  ~<_  A  ->  ( A  ~<_  ( B  +c  B
)  ->  om  ~<_  ( B  +c  B ) ) )
3229, 31syl 15 . . . . . . . . . . 11  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  ~<_  ( B  +c  B )  ->  om  ~<_  ( B  +c  B ) ) )
33 cdainf 7834 . . . . . . . . . . . . 13  |-  ( om  ~<_  B  <->  om  ~<_  ( B  +c  B ) )
3433biimpri 197 . . . . . . . . . . . 12  |-  ( om  ~<_  ( B  +c  B
)  ->  om  ~<_  B )
35 domrefg 6912 . . . . . . . . . . . . 13  |-  ( B  e.  dom  card  ->  B  ~<_  B )
36 infcdaabs 7848 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B  /\  B  ~<_  B )  ->  ( B  +c  B )  ~~  B )
37363com23 1157 . . . . . . . . . . . . . 14  |-  ( ( B  e.  dom  card  /\  B  ~<_  B  /\  om  ~<_  B )  ->  ( B  +c  B )  ~~  B )
38373expia 1153 . . . . . . . . . . . . 13  |-  ( ( B  e.  dom  card  /\  B  ~<_  B )  -> 
( om  ~<_  B  -> 
( B  +c  B
)  ~~  B )
)
3935, 38mpdan 649 . . . . . . . . . . . 12  |-  ( B  e.  dom  card  ->  ( om  ~<_  B  ->  ( B  +c  B )  ~~  B ) )
408, 34, 39syl2im 34 . . . . . . . . . . 11  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( om 
~<_  ( B  +c  B
)  ->  ( B  +c  B )  ~~  B
) )
4132, 40syld 40 . . . . . . . . . 10  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  ~<_  ( B  +c  B )  ->  ( B  +c  B )  ~~  B ) )
42 domen2 7020 . . . . . . . . . . 11  |-  ( ( B  +c  B ) 
~~  B  ->  ( A  ~<_  ( B  +c  B )  <->  A  ~<_  B ) )
4342biimpcd 215 . . . . . . . . . 10  |-  ( A  ~<_  ( B  +c  B
)  ->  ( ( B  +c  B )  ~~  B  ->  A  ~<_  B ) )
4441, 43sylcom 25 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  ~<_  ( B  +c  B )  ->  A  ~<_  B ) )
4528, 44syld 40 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( ( A  \  B )  +c  B
)  ~<_  ( B  +c  B )  ->  A  ~<_  B ) )
46 domnsym 7003 . . . . . . . 8  |-  ( A  ~<_  B  ->  -.  B  ~<  A )
4725, 45, 46syl56 30 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  ~<  B  ->  -.  B  ~<  A ) )
4822, 47mt2d 109 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  -.  ( A  \  B ) 
~<  B )
49 domtri2 7638 . . . . . . 7  |-  ( ( B  e.  dom  card  /\  ( A  \  B
)  e.  dom  card )  ->  ( B  ~<_  ( A  \  B )  <->  -.  ( A  \  B
)  ~<  B ) )
508, 16, 49syl2anc 642 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( B  ~<_  ( A  \  B )  <->  -.  ( A  \  B )  ~<  B ) )
5148, 50mpbird 223 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<_  ( A  \  B ) )
52 cdadom2 7829 . . . . 5  |-  ( B  ~<_  ( A  \  B
)  ->  ( ( A  \  B )  +c  B )  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
5351, 52syl 15 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  +c  B )  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
54 domtr 6930 . . . 4  |-  ( ( A  ~<_  ( ( A 
\  B )  +c  B )  /\  (
( A  \  B
)  +c  B )  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )  ->  A  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
5521, 53, 54syl2anc 642 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
56 domtr 6930 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )  ->  om  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
5729, 55, 56syl2anc 642 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  om  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
58 cdainf 7834 . . . . 5  |-  ( om  ~<_  ( A  \  B
)  <->  om  ~<_  ( ( A 
\  B )  +c  ( A  \  B
) ) )
5957, 58sylibr 203 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  om  ~<_  ( A 
\  B ) )
60 domrefg 6912 . . . . 5  |-  ( ( A  \  B )  e.  dom  card  ->  ( A  \  B )  ~<_  ( A  \  B
) )
6116, 60syl 15 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~<_  ( A  \  B ) )
62 infcdaabs 7848 . . . 4  |-  ( ( ( A  \  B
)  e.  dom  card  /\ 
om  ~<_  ( A  \  B )  /\  ( A  \  B )  ~<_  ( A  \  B ) )  ->  ( ( A  \  B )  +c  ( A  \  B
) )  ~~  ( A  \  B ) )
6316, 59, 61, 62syl3anc 1182 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  +c  ( A 
\  B ) ) 
~~  ( A  \  B ) )
64 domentr 6936 . . 3  |-  ( ( A  ~<_  ( ( A 
\  B )  +c  ( A  \  B
) )  /\  (
( A  \  B
)  +c  ( A 
\  B ) ) 
~~  ( A  \  B ) )  ->  A  ~<_  ( A  \  B ) )
6555, 63, 64syl2anc 642 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( A  \  B ) )
66 sbth 6997 . 2  |-  ( ( ( A  \  B
)  ~<_  A  /\  A  ~<_  ( A  \  B ) )  ->  ( A  \  B )  ~~  A
)
674, 65, 66syl2anc 642 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~~  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1696    \ cdif 3162    u. cun 3163    C_ wss 3165   class class class wbr 4039   omcom 4672   dom cdm 4705  (class class class)co 5874    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584    +c ccda 7809
This theorem is referenced by:  infdif2  7852  alephsuc3  8218  aleph1irr  12540
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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