Theorem cardonle 7237

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 7236	. 2 |- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )
2		enrefg 6809	. . 3 |- ( A e. On -> A ~~ A )
3		breq1 4032	. . . 4 |- ( x = A -> ( x ~~ A <-> A ~~ A ) )
4	3	intminss 3895	. . 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 3215	1 |- ( A e. On -> ( card ` A ) C_ A )

Syntax hints:   -> wi 4   e. wcel 2164   {crab 2476   C_ wss 3153   |^|cint 3870   class class class wbr 4029   Oncon0 4392   ` cfv 5246   ~~ cen 6783   cardccrd 7229

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-en 6786  df-card 7230

This theorem is referenced by:  card0  7238