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Theorem infdif 8045
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 957 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  e.  dom  card )
2 difss 3434 . . 3  |-  ( A 
\  B )  C_  A
3 ssdomg 7112 . . 3  |-  ( A  e.  dom  card  ->  ( ( A  \  B
)  C_  A  ->  ( A  \  B )  ~<_  A ) )
41, 2, 3ee10 1382 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~<_  A )
5 sdomdom 7094 . . . . . . . . 9  |-  ( B 
~<  A  ->  B  ~<_  A )
653ad2ant3 980 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<_  A )
7 numdom 7875 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\  B  ~<_  A )  ->  B  e.  dom  card )
81, 6, 7syl2anc 643 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  e.  dom  card )
9 unnum 8036 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  u.  B )  e.  dom  card )
101, 8, 9syl2anc 643 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  u.  B )  e.  dom  card )
11 ssun1 3470 . . . . . 6  |-  A  C_  ( A  u.  B
)
12 ssdomg 7112 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( A  C_  ( A  u.  B )  ->  A  ~<_  ( A  u.  B
) ) )
1310, 11, 12ee10 1382 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( A  u.  B
) )
14 undif1 3663 . . . . . 6  |-  ( ( A  \  B )  u.  B )  =  ( A  u.  B
)
15 ssnum 7876 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\  ( A  \  B
)  C_  A )  ->  ( A  \  B
)  e.  dom  card )
161, 2, 15sylancl 644 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  e. 
dom  card )
17 uncdadom 8007 . . . . . . 7  |-  ( ( ( A  \  B
)  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( A 
\  B )  u.  B )  ~<_  ( ( A  \  B )  +c  B ) )
1816, 8, 17syl2anc 643 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  u.  B )  ~<_  ( ( A  \  B )  +c  B
) )
1914, 18syl5eqbrr 4206 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  u.  B )  ~<_  ( ( A  \  B )  +c  B
) )
20 domtr 7119 . . . . 5  |-  ( ( A  ~<_  ( A  u.  B )  /\  ( A  u.  B )  ~<_  ( ( A  \  B )  +c  B
) )  ->  A  ~<_  ( ( A  \  B )  +c  B
) )
2113, 19, 20syl2anc 643 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( ( A  \  B )  +c  B
) )
22 simp3 959 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<  A )
23 sdomdom 7094 . . . . . . . . 9  |-  ( ( A  \  B ) 
~<  B  ->  ( A 
\  B )  ~<_  B )
24 cdadom1 8022 . . . . . . . . 9  |-  ( ( A  \  B )  ~<_  B  ->  ( ( A  \  B )  +c  B )  ~<_  ( B  +c  B ) )
2523, 24syl 16 . . . . . . . 8  |-  ( ( A  \  B ) 
~<  B  ->  ( ( A  \  B )  +c  B )  ~<_  ( B  +c  B ) )
26 domtr 7119 . . . . . . . . . . 11  |-  ( ( A  ~<_  ( ( A 
\  B )  +c  B )  /\  (
( A  \  B
)  +c  B )  ~<_  ( B  +c  B
) )  ->  A  ~<_  ( B  +c  B
) )
2726ex 424 . . . . . . . . . 10  |-  ( A  ~<_  ( ( A  \  B )  +c  B
)  ->  ( (
( A  \  B
)  +c  B )  ~<_  ( B  +c  B
)  ->  A  ~<_  ( B  +c  B ) ) )
2821, 27syl 16 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( ( A  \  B )  +c  B
)  ~<_  ( B  +c  B )  ->  A  ~<_  ( B  +c  B
) ) )
29 simp2 958 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  om  ~<_  A )
30 domtr 7119 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  ~<_  ( B  +c  B
) )  ->  om  ~<_  ( B  +c  B ) )
3130ex 424 . . . . . . . . . . . 12  |-  ( om  ~<_  A  ->  ( A  ~<_  ( B  +c  B
)  ->  om  ~<_  ( B  +c  B ) ) )
3229, 31syl 16 . . . . . . . . . . 11  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  ~<_  ( B  +c  B )  ->  om  ~<_  ( B  +c  B ) ) )
33 cdainf 8028 . . . . . . . . . . . . 13  |-  ( om  ~<_  B  <->  om  ~<_  ( B  +c  B ) )
3433biimpri 198 . . . . . . . . . . . 12  |-  ( om  ~<_  ( B  +c  B
)  ->  om  ~<_  B )
35 domrefg 7101 . . . . . . . . . . . . 13  |-  ( B  e.  dom  card  ->  B  ~<_  B )
36 infcdaabs 8042 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B  /\  B  ~<_  B )  ->  ( B  +c  B )  ~~  B )
37363com23 1159 . . . . . . . . . . . . . 14  |-  ( ( B  e.  dom  card  /\  B  ~<_  B  /\  om  ~<_  B )  ->  ( B  +c  B )  ~~  B )
38373expia 1155 . . . . . . . . . . . . 13  |-  ( ( B  e.  dom  card  /\  B  ~<_  B )  -> 
( om  ~<_  B  -> 
( B  +c  B
)  ~~  B )
)
3935, 38mpdan 650 . . . . . . . . . . . 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 7209 . . . . . . . . . . 11  |-  ( ( B  +c  B ) 
~~  B  ->  ( A  ~<_  ( B  +c  B )  <->  A  ~<_  B ) )
4342biimpcd 216 . . . . . . . . . 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 7192 . . . . . . . 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 7832 . . . . . . 7  |-  ( ( B  e.  dom  card  /\  ( A  \  B
)  e.  dom  card )  ->  ( B  ~<_  ( A  \  B )  <->  -.  ( A  \  B
)  ~<  B ) )
508, 16, 49syl2anc 643 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( B  ~<_  ( A  \  B )  <->  -.  ( A  \  B )  ~<  B ) )
5148, 50mpbird 224 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  B  ~<_  ( A  \  B ) )
52 cdadom2 8023 . . . . 5  |-  ( B  ~<_  ( A  \  B
)  ->  ( ( A  \  B )  +c  B )  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
5351, 52syl 16 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  +c  B )  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
54 domtr 7119 . . . 4  |-  ( ( A  ~<_  ( ( A 
\  B )  +c  B )  /\  (
( A  \  B
)  +c  B )  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )  ->  A  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
5521, 53, 54syl2anc 643 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
56 domtr 7119 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )  ->  om  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
5729, 55, 56syl2anc 643 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  om  ~<_  ( ( A  \  B )  +c  ( A  \  B ) ) )
58 cdainf 8028 . . . . 5  |-  ( om  ~<_  ( A  \  B
)  <->  om  ~<_  ( ( A 
\  B )  +c  ( A  \  B
) ) )
5957, 58sylibr 204 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  om  ~<_  ( A 
\  B ) )
60 domrefg 7101 . . . . 5  |-  ( ( A  \  B )  e.  dom  card  ->  ( A  \  B )  ~<_  ( A  \  B
) )
6116, 60syl 16 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B )  ~<_  ( A  \  B ) )
62 infcdaabs 8042 . . . 4  |-  ( ( ( A  \  B
)  e.  dom  card  /\ 
om  ~<_  ( A  \  B )  /\  ( A  \  B )  ~<_  ( A  \  B ) )  ->  ( ( A  \  B )  +c  ( A  \  B
) )  ~~  ( A  \  B ) )
6316, 59, 61, 62syl3anc 1184 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  (
( A  \  B
)  +c  ( A 
\  B ) ) 
~~  ( A  \  B ) )
64 domentr 7125 . . 3  |-  ( ( A  ~<_  ( ( A 
\  B )  +c  ( A  \  B
) )  /\  (
( A  \  B
)  +c  ( A 
\  B ) ) 
~~  ( A  \  B ) )  ->  A  ~<_  ( A  \  B ) )
6555, 63, 64syl2anc 643 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<  A )  ->  A  ~<_  ( A  \  B ) )
66 sbth 7186 . 2  |-  ( ( ( A  \  B
)  ~<_  A  /\  A  ~<_  ( A  \  B ) )  ->  ( A  \  B )  ~~  A
)
674, 65, 66syl2anc 643 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 177    /\ w3a 936    e. wcel 1721    \ cdif 3277    u. cun 3278    C_ wss 3280   class class class wbr 4172   omcom 4804   dom cdm 4837  (class class class)co 6040    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067   cardccrd 7778    +c ccda 8003
This theorem is referenced by:  infdif2  8046  alephsuc3  8411  aleph1irr  12800
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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