Theorem cardonle 7525

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 7524	. 2 |- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )
2		enrefg 7043	. . 3 |- ( A e. On -> A ~~ A )
3		breq1 4131	. . . 4 |- ( x = A -> ( x ~~ A <-> A ~~ A ) )
4	3	intminss 3993	. . 3 |- ( ( A e. On /\ A ~~ A ) -> |^| { x e. On | x ~~ A } C_ A )
5	2, 4	mpdan 425	. 2 |- ( A e. On -> |^| { x e. On | x ~~ A } C_ A )
6	1, 5	eqsstrd 3284	1 |- ( A e. On -> ( card ` A ) C_ A )

Syntax hints:   -> wi 4   e. wcel 2209   {crab 2532   C_ wss 3220   |^|cint 3968   class class class wbr 4128   Oncon0 4506   ` cfv 5375   ~~ cen 7013   cardccrd 7515

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-en 7016  df-card 7517

This theorem is referenced by:  card0  7526