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Theorem infdif 7830
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 960 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  e.  dom  card )
2 difss 3304 . . 3  |-  ( A 
\  B )  C_  A
3 ssdomg 6902 . . 3  |-  ( A  e.  dom  card  ->  ( ( A  \  B
)  C_  A  ->  ( A  \  B )  ~<_  A ) )
41, 2, 3ee10 1372 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~<_  A )
5 sdomdom 6884 . . . . . . . . 9  |-  ( B 
~<  A  ->  B  ~<_  A )
653ad2ant3 983 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<_  A )
7 numdom 7660 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\  B  ~<_  A )  ->  B  e.  dom  card )
81, 6, 7syl2anc 645 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  e.  dom  card )
9 unnum 7821 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  u.  B )  e.  dom  card )
101, 8, 9syl2anc 645 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  u.  B )  e.  dom  card )
11 ssun1 3339 . . . . . 6  |-  A  C_  ( A  u.  B
)
12 ssdomg 6902 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( A  C_  ( A  u.  B )  ->  A  ~<_  ( A  u.  B
) ) )
1310, 11, 12ee10 1372 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( A  u.  B
) )
14 undif1 3530 . . . . . 6  |-  ( ( A  \  B )  u.  B )  =  ( A  u.  B
)
15 ssnum 7661 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\  ( A  \  B
)  C_  A )  ->  ( A  \  B
)  e.  dom  card )
161, 2, 15sylancl 646 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  e. 
dom  card )
17 uncdadom 7792 . . . . . . 7  |-  ( ( ( A  \  B
)  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( A 
\  B )  u.  B )  ~<_  ( ( A  \  B )  +c  B ) )
1816, 8, 17syl2anc 645 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  u.  B )  ~<_  ( ( A  \  B )  +c  B
) )
1914, 18syl5eqbrr 4058 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  u.  B )  ~<_  ( ( A  \  B )  +c  B
) )
20 domtr 6909 . . . . 5  |-  ( ( A  ~<_  ( A  u.  B )  /\  ( A  u.  B )  ~<_  ( ( A  \  B )  +c  B
) )  ->  A  ~<_  ( ( A  \  B )  +c  B
) )
2113, 19, 20syl2anc 645 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( ( A  \  B )  +c  B
) )
22 simp3 962 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  A )
23 sdomdom 6884 . . . . . . . . 9  |-  ( ( A  \  B ) 
~<  B  ->  ( A 
\  B )  ~<_  B )
24 cdadom1 7807 . . . . . . . . 9  |-  ( ( A  \  B )  ~<_  B  ->  ( ( A  \  B )  +c  B )  ~<_  ( B  +c  B ) )
2523, 24syl 17 . . . . . . . 8  |-  ( ( A  \  B ) 
~<  B  ->  ( ( A  \  B )  +c  B )  ~<_  ( B  +c  B ) )
26 domtr 6909 . . . . . . . . . . 11  |-  ( ( A  ~<_  ( ( A 
\  B )  +c  B )  /\  (
( A  \  B
)  +c  B )  ~<_  ( B  +c  B
) )  ->  A  ~<_  ( B  +c  B
) )
2726ex 425 . . . . . . . . . 10  |-  ( A  ~<_  ( ( A  \  B )  +c  B
)  ->  ( (
( A  \  B
)  +c  B )  ~<_  ( B  +c  B
)  ->  A  ~<_  ( B  +c  B ) ) )
2821, 27syl 17 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( ( A  \  B )  +c  B
)  ~<_  ( B  +c  B )  ->  A  ~<_  ( B  +c  B
) ) )
29 simp2 961 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  om  ~<_  A )
30 domtr 6909 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  ~<_  ( B  +c  B
) )  ->  om  ~<_  ( B  +c  B ) )
3130ex 425 . . . . . . . . . . . 12  |-  ( om  ~<_  A  ->  ( A  ~<_  ( B  +c  B
)  ->  om  ~<_  ( B  +c  B ) ) )
3229, 31syl 17 . . . . . . . . . . 11  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  ~<_  ( B  +c  B )  ->  om  ~<_  ( B  +c  B ) ) )
33 cdainf 7813 . . . . . . . . . . . . 13  |-  ( om  ~<_  B  <->  om  ~<_  ( B  +c  B ) )
3433biimpri 199 . . . . . . . . . . . 12  |-  ( om  ~<_  ( B  +c  B
)  ->  om  ~<_  B )
35 domrefg 6891 . . . . . . . . . . . . 13  |-  ( B  e.  dom  card  ->  B  ~<_  B )
36 infcdaabs 7827 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B  /\  B  ~<_  B )  ->  ( B  +c  B )  ~~  B )
37363com23 1162 . . . . . . . . . . . . . 14  |-  ( ( B  e.  dom  card  /\  B  ~<_  B  /\  om  ~<_  B )  ->  ( B  +c  B )  ~~  B )
38373expia 1158 . . . . . . . . . . . . 13  |-  ( ( B  e.  dom  card  /\  B  ~<_  B )  -> 
( om  ~<_  B  -> 
( B  +c  B
)  ~~  B )
)
3935, 38mpdan 652 . . . . . . . . . . . 12  |-  ( B  e.  dom  card  ->  ( om  ~<_  B  ->  ( B  +c  B )  ~~  B ) )
408, 34, 39syl2im 36 . . . . . . . . . . 11  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( om 
~<_  ( B  +c  B
)  ->  ( B  +c  B )  ~~  B
) )
4132, 40syld 42 . . . . . . . . . 10  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  ~<_  ( B  +c  B )  ->  ( B  +c  B )  ~~  B ) )
42 domen2 6999 . . . . . . . . . . 11  |-  ( ( B  +c  B ) 
~~  B  ->  ( A  ~<_  ( B  +c  B )  <->  A  ~<_  B ) )
4342biimpcd 217 . . . . . . . . . 10  |-  ( A  ~<_  ( B  +c  B
)  ->  ( ( B  +c  B )  ~~  B  ->  A  ~<_  B ) )
4441, 43sylcom 27 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  ~<_  ( B  +c  B )  ->  A  ~<_  B ) )
4528, 44syld 42 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( ( A  \  B )  +c  B
)  ~<_  ( B  +c  B )  ->  A  ~<_  B ) )
46 domnsym 6982 . . . . . . . 8  |-  ( A  ~<_  B  ->  -.  B  ~<  A )
4725, 45, 46syl56 32 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  ~<  B  ->  -.  B  ~<  A ) )
4822, 47mt2d 111 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  -.  ( A  \  B ) 
~<  B )
49 domtri2 7617 . . . . . . 7  |-  ( ( B  e.  dom  card  /\  ( A  \  B
)  e.  dom  card )  ->  ( B  ~<_  ( A  \  B )  <->  -.  ( A  \  B
)  ~<  B ) )
508, 16, 49syl2anc 645 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( B  ~<_  ( A  \  B )  <->  -.  ( A  \  B )  ~<  B ) )
5148, 50mpbird 225 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<_  ( A  \  B ) )
52 cdadom2 7808 . . . . 5  |-  ( B  ~<_  ( A  \  B
)  ->  ( ( A  \  B )  +c  B )  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
5351, 52syl 17 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  +c  B )  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
54 domtr 6909 . . . 4  |-  ( ( A  ~<_  ( ( A 
\  B )  +c  B )  /\  (
( A  \  B
)  +c  B )  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )  ->  A  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
5521, 53, 54syl2anc 645 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
56 domtr 6909 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )  ->  om  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
5729, 55, 56syl2anc 645 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  om  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
58 cdainf 7813 . . . . 5  |-  ( om  ~<_  ( A  \  B
)  <->  om  ~<_  ( ( A 
\  B )  +c  ( A  \  B
) ) )
5957, 58sylibr 205 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  om  ~<_  ( A 
\  B ) )
60 domrefg 6891 . . . . 5  |-  ( ( A  \  B )  e.  dom  card  ->  ( A  \  B )  ~<_  ( A  \  B
) )
6116, 60syl 17 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~<_  ( A  \  B ) )
62 infcdaabs 7827 . . . 4  |-  ( ( ( A  \  B
)  e.  dom  card  /\ 
om  ~<_  ( A  \  B )  /\  ( A  \  B )  ~<_  ( A  \  B ) )  ->  ( ( A  \  B )  +c  ( A  \  B
) )  ~~  ( A  \  B ) )
6316, 59, 61, 62syl3anc 1187 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  +c  ( A 
\  B ) ) 
~~  ( A  \  B ) )
64 domentr 6915 . . 3  |-  ( ( A  ~<_  ( ( A 
\  B )  +c  ( A  \  B
) )  /\  (
( A  \  B
)  +c  ( A 
\  B ) ) 
~~  ( A  \  B ) )  ->  A  ~<_  ( A  \  B ) )
6555, 63, 64syl2anc 645 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( A  \  B ) )
66 sbth 6976 . 2  |-  ( ( ( A  \  B
)  ~<_  A  /\  A  ~<_  ( A  \  B ) )  ->  ( A  \  B )  ~~  A
)
674, 65, 66syl2anc 645 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~~  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ w3a 939    e. wcel 1688    \ cdif 3150    u. cun 3151    C_ wss 3153   class class class wbr 4024   omcom 4655   dom cdm 4688  (class class class)co 5819    ~~ cen 6855    ~<_ cdom 6856    ~< csdm 6857   cardccrd 7563    +c ccda 7788
This theorem is referenced by:  infdif2  7831  alephsuc3  8197  aleph1irr  12518
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-oi 7220  df-card 7567  df-cda 7789
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