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Theorem domtri2 8387
Description: Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
domtri2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )

Proof of Theorem domtri2
StepHypRef Expression
1 carddom2 8375 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
2 cardon 8342 . . . 4  |-  ( card `  A )  e.  On
3 cardon 8342 . . . 4  |-  ( card `  B )  e.  On
4 ontri1 4921 . . . 4  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
52, 3, 4mp2an 672 . . 3  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
6 cardsdom2 8386 . . . . 5  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  B )  e.  (
card `  A )  <->  B 
~<  A ) )
76ancoms 453 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  e.  (
card `  A )  <->  B 
~<  A ) )
87notbid 294 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( -.  ( card `  B )  e.  ( card `  A
)  <->  -.  B  ~<  A ) )
95, 8syl5bb 257 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  -.  B  ~<  A ) )
101, 9bitr3d 255 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1819    C_ wss 3471   class class class wbr 4456   Oncon0 4887   dom cdm 5008   ` cfv 5594    ~<_ cdom 7533    ~< csdm 7534   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-card 8337
This theorem is referenced by:  fidomtri  8391  harsdom  8393  infdif  8606  infdif2  8607  infunsdom1  8610  infunsdom  8611  infxp  8612  domtri  8948  canthp1lem2  9048  pwfseqlem4a  9056  pwfseqlem4  9057  gchaleph  9066  numinfctb  31214
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