Theorem cardonle 8783

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 8781	. 2 |- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )
2		enrefg 7987	. . 3 |- ( A e. On -> A ~~ A )
3		breq1 4656	. . . 4 |- ( x = A -> ( x ~~ A <-> A ~~ A ) )
4	3	intminss 4503	. . 3 |- ( ( A e. On /\ A ~~ A ) -> |^| { x e. On | x ~~ A } C_ A )
5	2, 4	mpdan 702	. 2 |- ( A e. On -> |^| { x e. On | x ~~ A } C_ A )
6	1, 5	eqsstrd 3639	1 |- ( A e. On -> ( card ` A ) C_ A )

Syntax hints:   -> wi 4   e. wcel 1990   {crab 2916   C_ wss 3574   |^|cint 4475   class class class wbr 4653   Oncon0 5723   ` cfv 5888   ~~ cen 7952   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956  df-card 8765

This theorem is referenced by:  card0  8784  iscard  8801  iscard2  8802  carduni  8807  cardom  8812  alephinit  8918  cfle  9076  cfflb  9081  pwfseqlem5  9485