Theorem rankonid 4667

Description: The rank of an ordinal number is itself. Proposition 9.18 of [TakeutiZaring] p. 79 and its converse.

Assertion

Ref	Expression
rankonid	|- (A e. On <-> (rank` A) = A)

Proof of Theorem rankonid

Step	Hyp	Ref	Expression
1		fveq2 3709	. . . 4 |- (x = y -> (rank` x) = (rank` y))
2		id 59	. . . 4 |- (x = y -> x = y)
3	1, 2	eqeq12d 1481	. . 3 |- (x = y -> ((rank` x) = x <-> (rank` y) = y))
4		fveq2 3709	. . . 4 |- (x = A -> (rank` x) = (rank` A))
5		id 59	. . . 4 |- (x = A -> x = A)
6	4, 5	eqeq12d 1481	. . 3 |- (x = A -> ((rank` x) = x <-> (rank` A) = A))
7		eleq1 1526	. . . . . . . . . . 11 |- ((rank` y) = y -> ((rank` y) e. z <-> y e. z))
8	7	r19.20si 1698	. . . . . . . . . 10 |- (A.y e. x (rank` y) = y -> A.y e. x ((rank` y) e. z <-> y e. z))
9		r19.15 1745	. . . . . . . . . 10 |- (A.y e. x ((rank` y) e. z <-> y e. z) -> (A.y e. x (rank` y) e. z <-> A.y e. x y e. z))
10	8, 9	syl 10	. . . . . . . . 9 |- (A.y e. x (rank` y) = y -> (A.y e. x (rank` y) e. z <-> A.y e. x y e. z))
11		dfss3 2049	. . . . . . . . 9 |- (x (_ z <-> A.y e. x y e. z)
12	10, 11	syl6bbr 536	. . . . . . . 8 |- (A.y e. x (rank` y) = y -> (A.y e. x (rank` y) e. z <-> x (_ z))
13	12	rabbisdv 1798	. . . . . . 7 |- (A.y e. x (rank` y) = y -> {z e. On | A.y e. x (rank` y) e. z} = {z e. On | x (_ z})
14	13	inteqd 2528	. . . . . 6 |- (A.y e. x (rank` y) = y -> |^|{z e. On | A.y e. x (rank` y) e. z} = |^|{z e. On | x (_ z})
15		visset 1804	. . . . . . 7 |- x e. V
16	15	rankval3 4653	. . . . . 6 |- (rank` x) = |^|{z e. On | A.y e. x (rank` y) e. z}
17	14, 16	syl5eq 1511	. . . . 5 |- (A.y e. x (rank` y) = y -> (rank` x) = |^|{z e. On | x (_ z})
18		intmin 2543	. . . . 5 |- (x e. On -> |^|{z e. On | x (_ z} = x)
19	17, 18	sylan9eqr 1521	. . . 4 |- ((x e. On /\ A.y e. x (rank` y) = y) -> (rank` x) = x)
20	19	ex 373	. . 3 |- (x e. On -> (A.y e. x (rank` y) = y -> (rank` x) = x))
21	3, 6, 20	tfis3 3120	. 2 |- (A e. On -> (rank` A) = A)
22		rankon 4643	. . 3 |- (rank` A) e. On
23		eleq1 1526	. . 3 |- ((rank` A) = A -> ((rank` A) e. On <-> A e. On))
24	22, 23	mpbii 193	. 2 |- ((rank` A) = A -> A e. On)
25	21, 24	impbi 157	1 |- (A e. On <-> (rank` A) = A)

Syntax hints:   <-> wb 146   = wceq 953   e. wcel 955  A.wral 1637  {crab 1640   (_ wss 2037  |^|cint 2523  Oncon0 2938  ` cfv 3172  rankcrnk 4614

This theorem is referenced by:  rankeq0 4668  rankr1id 4669

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-r1 4615  df-rank 4616

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