Theorem rankonid 4675

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 3715	. . . 4 |- (x = y -> (rank` x) = (rank` y))
2		id 59	. . . 4 |- (x = y -> x = y)
3	1, 2	eqeq12d 1486	. . 3 |- (x = y -> ((rank` x) = x <-> (rank` y) = y))
4		fveq2 3715	. . . 4 |- (x = A -> (rank` x) = (rank` A))
5		id 59	. . . 4 |- (x = A -> x = A)
6	4, 5	eqeq12d 1486	. . 3 |- (x = A -> ((rank` x) = x <-> (rank` A) = A))
7		eleq1 1531	. . . . . . . . . . 11 |- ((rank` y) = y -> ((rank` y) e. z <-> y e. z))
8	7	r19.20si 1703	. . . . . . . . . 10 |- (A.y e. x (rank` y) = y -> A.y e. x ((rank` y) e. z <-> y e. z))
9		r19.15 1750	. . . . . . . . . 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 2055	. . . . . . . . 9 |- (x (_ z <-> A.y e. x y e. z)
12	10, 11	syl6bbr 537	. . . . . . . 8 |- (A.y e. x (rank` y) = y -> (A.y e. x (rank` y) e. z <-> x (_ z))
13	12	rabbisdv 1803	. . . . . . 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 2533	. . . . . 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 1809	. . . . . . 7 |- x e. V
16	15	rankval3 4661	. . . . . 6 |- (rank` x) = |^|{z e. On | A.y e. x (rank` y) e. z}
17	14, 16	syl5eq 1516	. . . . 5 |- (A.y e. x (rank` y) = y -> (rank` x) = |^|{z e. On | x (_ z})
18		intmin 2548	. . . . 5 |- (x e. On -> |^|{z e. On | x (_ z} = x)
19	17, 18	sylan9eqr 1526	. . . 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 3125	. 2 |- (A e. On -> (rank` A) = A)
22		rankon 4651	. . 3 |- (rank` A) e. On
23		eleq1 1531	. . 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 954   e. wcel 956  A.wral 1642  {crab 1645   (_ wss 2043  |^|cint 2528  Oncon0 2943  ` cfv 3177  rankcrnk 4622

This theorem is referenced by:  rankeq0 4676  rankr1id 4677

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623  df-rank 4624

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