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Theorem carddom2 8354
Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8925, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
carddom2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )

Proof of Theorem carddom2
StepHypRef Expression
1 carddomi2 8347 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )
2 brdom2 7542 . . 3  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
3 cardon 8321 . . . . . . . 8  |-  ( card `  A )  e.  On
43onelssi 4986 . . . . . . 7  |-  ( (
card `  B )  e.  ( card `  A
)  ->  ( card `  B )  C_  ( card `  A ) )
5 carddomi2 8347 . . . . . . . 8  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  ->  B  ~<_  A ) )
65ancoms 453 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  ->  B  ~<_  A ) )
7 domnsym 7640 . . . . . . 7  |-  ( B  ~<_  A  ->  -.  A  ~<  B )
84, 6, 7syl56 34 . . . . . 6  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  e.  (
card `  A )  ->  -.  A  ~<  B ) )
98con2d 115 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<  B  ->  -.  ( card `  B )  e.  (
card `  A )
) )
10 cardon 8321 . . . . . 6  |-  ( card `  B )  e.  On
11 ontri1 4912 . . . . . 6  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
123, 10, 11mp2an 672 . . . . 5  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
139, 12syl6ibr 227 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<  B  ->  ( card `  A
)  C_  ( card `  B ) ) )
14 carden2b 8344 . . . . . 6  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
15 eqimss 3556 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( card `  A )  C_  ( card `  B ) )
1614, 15syl 16 . . . . 5  |-  ( A 
~~  B  ->  ( card `  A )  C_  ( card `  B )
)
1716a1i 11 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~~  B  ->  ( card `  A
)  C_  ( card `  B ) ) )
1813, 17jaod 380 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( A 
~<  B  \/  A  ~~  B )  ->  ( card `  A )  C_  ( card `  B )
) )
192, 18syl5bi 217 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  ->  ( card `  A
)  C_  ( card `  B ) ) )
201, 19impbid 191 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ` cfv 5586    ~~ cen 7510    ~<_ cdom 7511    ~< csdm 7512   cardccrd 8312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-card 8316
This theorem is referenced by:  carduni  8358  carden2  8364  cardsdom2  8365  domtri2  8366  infxpidm2  8390  cardaleph  8466  infenaleph  8468  alephinit  8472  ficardun2  8579  ackbij2  8619  cfflb  8635  fin1a2lem9  8784  carddom  8925  pwfseqlem5  9037  hashdom  12409
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