Theorem rankonid 5806

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 4681	. . . 4 |- (x = y -> (rank` x) = (rank` y))
2		id 73	. . . 4 |- (x = y -> x = y)
3	1, 2	eqeq12d 1899	. . 3 |- (x = y -> ((rank` x) = x <-> (rank` y) = y))
4		fveq2 4681	. . . 4 |- (x = A -> (rank` x) = (rank` A))
5		id 73	. . . 4 |- (x = A -> x = A)
6	4, 5	eqeq12d 1899	. . 3 |- (x = A -> ((rank` x) = x <-> (rank` A) = A))
7		eleq1 1957	. . . . . . . . . . 11 |- ((rank` y) = y -> ((rank` y) e. z <-> y e. z))
8	7	ralimi 2168	. . . . . . . . . 10 |- (A.y e. x (rank` y) = y -> A.y e. x ((rank` y) e. z <-> y e. z))
9		ralbi 2223	. . . . . . . . . 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 12	. . . . . . . . 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 2611	. . . . . . . . 9 |- (x C_ z <-> A.y e. x y e. z)
12	10, 11	syl6bbr 597	. . . . . . . 8 |- (A.y e. x (rank` y) = y -> (A.y e. x (rank` y) e. z <-> x C_ z))
13	12	rabbidv 2287	. . . . . . 7 |- (A.y e. x (rank` y) = y -> {z e. On | A.y e. x (rank` y) e. z} = {z e. On | x C_ z})
14	13	inteqd 3219	. . . . . 6 |- (A.y e. x (rank` y) = y -> |^|{z e. On | A.y e. x (rank` y) e. z} = |^|{z e. On | x C_ z})
15		visset 2295	. . . . . . 7 |- x e. _V
16	15	rankval3 5792	. . . . . 6 |- (rank` x) = |^|{z e. On | A.y e. x (rank` y) e. z}
17	14, 16	syl5eq 1940	. . . . 5 |- (A.y e. x (rank` y) = y -> (rank` x) = |^|{z e. On | x C_ z})
18		intmin 3237	. . . . 5 |- (x e. On -> |^|{z e. On | x C_ z} = x)
19	17, 18	sylan9eqr 1951	. . . 4 |- ((x e. On /\ A.y e. x (rank` y) = y) -> (rank` x) = x)
20	19	ex 402	. . 3 |- (x e. On -> (A.y e. x (rank` y) = y -> (rank` x) = x))
21	3, 6, 20	tfis3 3941	. 2 |- (A e. On -> (rank` A) = A)
22		rankon 5782	. . 3 |- (rank` A) e. On
23		eleq1 1957	. . 3 |- ((rank` A) = A -> ((rank` A) e. On <-> A e. On))
24	22, 23	mpbii 210	. 2 |- ((rank` A) = A -> A e. On)
25	21, 24	impbii 174	1 |- (A e. On <-> (rank` A) = A)

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108   C_ wss 2593  |^|cint 3214  Oncon0 3657  ` cfv 3998  rankcrnk 5749

This theorem is referenced by:  rankeq0 5807  rankr1id 5808  onsubcum 14442

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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

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-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  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-lim 3662  df-suc 3663  df-om 3950  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-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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