Theorem cardonle 7143

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 7142	. 2 |- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )
2		enrefg 6730	. . 3 |- ( A e. On -> A ~~ A )
3		breq1 3985	. . . 4 |- ( x = A -> ( x ~~ A <-> A ~~ A ) )
4	3	intminss 3849	. . 3 |- ( ( A e. On /\ A ~~ A ) -> |^| { x e. On | x ~~ A } C_ A )
5	2, 4	mpdan 418	. 2 |- ( A e. On -> |^| { x e. On | x ~~ A } C_ A )
6	1, 5	eqsstrd 3178	1 |- ( A e. On -> ( card ` A ) C_ A )

Syntax hints:   -> wi 4   e. wcel 2136   {crab 2448   C_ wss 3116   |^|cint 3824   class class class wbr 3982   Oncon0 4341   ` cfv 5188   ~~ cen 6704   cardccrd 7135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-en 6707  df-card 7136

This theorem is referenced by:  card0  7144