Theorem cardonle 4832

Description: The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85.

Assertion

Ref	Expression
cardonle	|- (A e. On -> (card` A) (_ A)

Proof of Theorem cardonle

Step	Hyp	Ref	Expression
1		oncardval 4829	. 2 |- (A e. On -> (card` A) = |^|{x e. On | x ~~ A})
2		enrefg 4396	. . 3 |- (A e. On -> A ~~ A)
3		breq1 2627	. . . . 5 |- (x = A -> (x ~~ A <-> A ~~ A))
4	3	elrab 1908	. . . 4 |- (A e. {x e. On | x ~~ A} <-> (A e. On /\ A ~~ A))
5		intss1 2552	. . . 4 |- (A e. {x e. On | x ~~ A} -> |^|{x e. On | x ~~ A} (_ A)
6	4, 5	sylbir 201	. . 3 |- ((A e. On /\ A ~~ A) -> |^|{x e. On | x ~~ A} (_ A)
7	2, 6	mpdan 706	. 2 |- (A e. On -> |^|{x e. On | x ~~ A} (_ A)
8	1, 7	eqsstrd 2098	1 |- (A e. On -> (card` A) (_ A)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  {crab 1651   (_ wss 2050  |^|cint 2537   class class class wbr 2624  Oncon0 2954  ` cfv 3188   ~~ cen 4370  cardccrd 4823

This theorem is referenced by:  card0 4833  cardnn 4834  cardom 4835  oncard 4839  iscard 4864  iscard2 4865  carduni 4869  cfle 4925

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-en 4374  df-card 4826

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