Theorem oncardid 5867

Description: Any ordinal number is equinumerous to its cardinal number. Unlike cardid 5977, this theorem does not require the Axiom of Choice.

Assertion

Ref	Expression
oncardid	|- (A e. On -> (card` A) ~~ A)

Proof of Theorem oncardid

Step	Hyp	Ref	Expression
1		oncardval 5865	. . 3 |- (A e. On -> (card` A) = |^|{x e. On | x ~~ A})
2		onint 3876	. . . 4 |- (({x e. On | x ~~ A} C_ On /\ {x e. On | x ~~ A} =/= (/)) -> |^|{x e. On | x ~~ A} e. {x e. On | x ~~ A})
3		ssrab2 2692	. . . 4 |- {x e. On | x ~~ A} C_ On
4		fvex 4689	. . . . . 6 |- (card` A) e. _V
5	1, 4	syl6eqelr 1980	. . . . 5 |- (A e. On -> |^|{x e. On | x ~~ A} e. _V)
6		intex 3465	. . . . 5 |- ({x e. On | x ~~ A} =/= (/) <-> |^|{x e. On | x ~~ A} e. _V)
7	5, 6	sylibr 217	. . . 4 |- (A e. On -> {x e. On | x ~~ A} =/= (/))
8	2, 3, 7	sylancr 526	. . 3 |- (A e. On -> |^|{x e. On | x ~~ A} e. {x e. On | x ~~ A})
9	1, 8	eqeltrd 1971	. 2 |- (A e. On -> (card` A) e. {x e. On | x ~~ A})
10		breq1 3341	. . . 4 |- (x = (card` A) -> (x ~~ A <-> (card` A) ~~ A))
11	10	elrab 2414	. . 3 |- ((card` A) e. {x e. On | x ~~ A} <-> ((card` A) e. On /\ (card` A) ~~ A))
12	11	simprbi 353	. 2 |- ((card` A) e. {x e. On | x ~~ A} -> (card` A) ~~ A)
13	9, 12	syl 12	1 |- (A e. On -> (card` A) ~~ A)

Syntax hints:   -> wi 3   e. wcel 1300   =/= wne 2017  {crab 2108  _Vcvv 2292   C_ wss 2593  (/)c0 2875  |^|cint 3214   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~~ cen 5423  cardccrd 5859

This theorem is referenced by:  cardnn 5870  cardom 5872

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-en 5427  df-card 5862

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