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Theorem cardonle 7779
Description: The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
cardonle  |-  ( A  e.  On  ->  ( card `  A )  C_  A )

Proof of Theorem cardonle
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oncardval 7777 . 2  |-  ( A  e.  On  ->  ( card `  A )  = 
|^| { x  e.  On  |  x  ~~  A }
)
2 enrefg 7077 . . 3  |-  ( A  e.  On  ->  A  ~~  A )
3 breq1 4158 . . . 4  |-  ( x  =  A  ->  (
x  ~~  A  <->  A  ~~  A ) )
43intminss 4020 . . 3  |-  ( ( A  e.  On  /\  A  ~~  A )  ->  |^| { x  e.  On  |  x  ~~  A }  C_  A )
52, 4mpdan 650 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  x  ~~  A }  C_  A )
61, 5eqsstrd 3327 1  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   {crab 2655    C_ wss 3265   |^|cint 3994   class class class wbr 4155   Oncon0 4524   ` cfv 5396    ~~ cen 7044   cardccrd 7757
This theorem is referenced by:  card0  7780  iscard  7797  iscard2  7798  carduni  7803  cardom  7808  alephinit  7911  cfle  8069  cfflb  8074  pwfseqlem5  8473
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-en 7048  df-card 7761
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