Theorem cardonle 7485

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.

Step	Hyp	Ref	Expression
1		oncardval 7484	. 2 |- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )
2		enrefg 7005	. . 3 |- ( A e. On -> A ~~ A )
3		breq1 4114	. . . 4 |- ( x = A -> ( x ~~ A <-> A ~~ A ) )
4	3	intminss 3976	. . 3 |- ( ( A e. On /\ A ~~ A ) -> |^| { x e. On | x ~~ A } C_ A )
5	2, 4	mpdan 421	. 2 |- ( A e. On -> |^| { x e. On | x ~~ A } C_ A )
6	1, 5	eqsstrd 3276	1 |- ( A e. On -> ( card ` A ) C_ A )

Syntax hints:   -> wi 4   e. wcel 2205   {crab 2526   C_ wss 3213   |^|cint 3951   class class class wbr 4111   Oncon0 4486   ` cfv 5354   ~~ cen 6975   cardccrd 7475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-en 6978  df-card 7477

This theorem is referenced by:  card0  7486