Theorem oncardid 4808

Description: Any ordinal number is equinumerous to its cardinal number. Unlike cardid 4815, 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 4806	. . 3 |- (A e. On -> (card` A) = |^|{x e. On | x ~~ A})
2		fvex 3729	. . . . . 6 |- (card` A) e. V
3	1, 2	syl6eqelr 1556	. . . . 5 |- (A e. On -> |^|{x e. On | x ~~ A} e. V)
4		intex 2726	. . . . 5 |- ({x e. On | x ~~ A} =/= (/) <-> |^|{x e. On | x ~~ A} e. V)
5	3, 4	sylibr 200	. . . 4 |- (A e. On -> {x e. On | x ~~ A} =/= (/))
6		ssrab2 2129	. . . . 5 |- {x e. On | x ~~ A} (_ On
7		onint 3003	. . . . 5 |- (({x e. On | x ~~ A} (_ On /\ {x e. On | x ~~ A} =/= (/)) -> |^|{x e. On | x ~~ A} e. {x e. On | x ~~ A})
8	6, 7	mpan 694	. . . 4 |- ({x e. On | x ~~ A} =/= (/) -> |^|{x e. On | x ~~ A} e. {x e. On | x ~~ A})
9	5, 8	syl 10	. . 3 |- (A e. On -> |^|{x e. On | x ~~ A} e. {x e. On | x ~~ A})
10	1, 9	eqeltrd 1547	. 2 |- (A e. On -> (card` A) e. {x e. On | x ~~ A})
11		breq1 2619	. . . 4 |- (x = (card` A) -> (x ~~ A <-> (card` A) ~~ A))
12	11	elrab 1903	. . 3 |- ((card` A) e. {x e. On | x ~~ A} <-> ((card` A) e. On /\ (card` A) ~~ A))
13	12	pm3.27bi 326	. 2 |- ((card` A) e. {x e. On | x ~~ A} -> (card` A) ~~ A)
14	10, 13	syl 10	1 |- (A e. On -> (card` A) ~~ A)

Syntax hints:   -> wi 3   e. wcel 957   =/= wne 1584  {crab 1647  Vcvv 1809   (_ wss 2045  (/)c0 2278  |^|cint 2530   class class class wbr 2616  Oncon0 2945  ` cfv 3179   ~~ cen 4361  cardccrd 4800

This theorem is referenced by:  cardnn 4811  cardom 4812

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195  df-en 4364  df-card 4803

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