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Theorem cardonle 8370
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 8368 . 2  |-  ( A  e.  On  ->  ( card `  A )  = 
|^| { x  e.  On  |  x  ~~  A }
)
2 enrefg 7585 . . 3  |-  ( A  e.  On  ->  A  ~~  A )
3 breq1 4398 . . . 4  |-  ( x  =  A  ->  (
x  ~~  A  <->  A  ~~  A ) )
43intminss 4254 . . 3  |-  ( ( A  e.  On  /\  A  ~~  A )  ->  |^| { x  e.  On  |  x  ~~  A }  C_  A )
52, 4mpdan 666 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  x  ~~  A }  C_  A )
61, 5eqsstrd 3476 1  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   {crab 2758    C_ wss 3414   |^|cint 4227   class class class wbr 4395   Oncon0 5410   ` cfv 5569    ~~ cen 7551   cardccrd 8348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-en 7555  df-card 8352
This theorem is referenced by:  card0  8371  iscard  8388  iscard2  8389  carduni  8394  cardom  8399  alephinit  8508  cfle  8666  cfflb  8671  pwfseqlem5  9071
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