Theorem oncardval 4829

Description: The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 4836, this theorem does not require the Axiom of Choice.

Assertion

Ref	Expression
oncardval	|- (A e. On -> (card` A) = |^|{x e. On | x ~~ A})

Distinct variable group:   x,A

Proof of Theorem oncardval

Step	Hyp	Ref	Expression
1		enrefg 4396	. . . . . 6 |- (A e. On -> A ~~ A)
2		breq1 2627	. . . . . . 7 |- (x = A -> (x ~~ A <-> A ~~ A))
3	2	rcla4ev 1880	. . . . . 6 |- ((A e. On /\ A ~~ A) -> E.x e. On x ~~ A)
4	1, 3	mpdan 706	. . . . 5 |- (A e. On -> E.x e. On x ~~ A)
5		rabn0 2296	. . . . 5 |- ({x e. On | x ~~ A} =/= (/) <-> E.x e. On x ~~ A)
6	4, 5	sylibr 200	. . . 4 |- (A e. On -> {x e. On | x ~~ A} =/= (/))
7		ssrab2 2134	. . . . 5 |- {x e. On | x ~~ A} (_ On
8		oninton 3018	. . . . 5 |- (({x e. On | x ~~ A} (_ On /\ {x e. On | x ~~ A} =/= (/)) -> |^|{x e. On | x ~~ A} e. On)
9	7, 8	mpan 697	. . . 4 |- ({x e. On | x ~~ A} =/= (/) -> |^|{x e. On | x ~~ A} e. On)
10	6, 9	syl 10	. . 3 |- (A e. On -> |^|{x e. On | x ~~ A} e. On)
11		breq2 2628	. . . . . 6 |- (y = A -> (x ~~ y <-> x ~~ A))
12	11	rabbisdv 1810	. . . . 5 |- (y = A -> {x e. On | x ~~ y} = {x e. On | x ~~ A})
13	12	inteqd 2542	. . . 4 |- (y = A -> |^|{x e. On | x ~~ y} = |^|{x e. On | x ~~ A})
14	13	fvopabg 3791	. . 3 |- ((A e. On /\ |^|{x e. On | x ~~ A} e. On) -> ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A) = |^|{x e. On | x ~~ A})
15	10, 14	mpdan 706	. 2 |- (A e. On -> ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A) = |^|{x e. On | x ~~ A})
16		df-card 4826	. . 3 |- card = {<.y, z>. | z = |^|{x e. On | x ~~ y}}
17	16	fveq1i 3731	. 2 |- (card` A) = ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A)
18	15, 17	syl5eq 1522	1 |- (A e. On -> (card` A) = |^|{x e. On | x ~~ A})

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960   =/= wne 1588  E.wrex 1649  {crab 1651   (_ wss 2050  (/)c0 2283  |^|cint 2537   class class class wbr 2624  {copab 2671  Oncon0 2954  ` cfv 3188   ~~ cen 4370  cardccrd 4823

This theorem is referenced by:  oncardon 4830  oncardid 4831  cardonle 4832

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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