Theorem rankonid 3538

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

Assertion

Ref	Expression
rankonid	⊢ (A ∈ On ↔ (rank ‘A) = A)

Proof of Theorem rankonid

Step	Hyp	Ref	Expression
1		fveq2 2832	. . . 4 ⊢ (x = y → (rank ‘x) = (rank ‘y))
2		id 9	. . . 4 ⊢ (x = y → x = y)
3	1, 2	cleq12d 1115	. . 3 ⊢ (x = y → ((rank ‘x) = x ↔ (rank ‘y) = y))
4		fveq2 2832	. . . 4 ⊢ (x = A → (rank ‘x) = (rank ‘A))
5		id 9	. . . 4 ⊢ (x = A → x = A)
6	4, 5	cleq12d 1115	. . 3 ⊢ (x = A → ((rank ‘x) = x ↔ (rank ‘A) = A))
7		eleq1 1149	. . . . . . . . . . 11 ⊢ ((rank ‘y) = y → ((rank ‘y) ∈ z ↔ y ∈ z))
8	7	r19.20si 1254	. . . . . . . . . 10 ⊢ (∀y ∈ x (rank ‘y) = y → ∀y ∈ x ((rank ‘y) ∈ z ↔ y ∈ z))
9		r19.15 1292	. . . . . . . . . 10 ⊢ (∀y ∈ x ((rank ‘y) ∈ z ↔ y ∈ z) → (∀y ∈ x (rank ‘y) ∈ z ↔ ∀y ∈ x y ∈ z))
10	8, 9	syl 12	. . . . . . . . 9 ⊢ (∀y ∈ x (rank ‘y) = y → (∀y ∈ x (rank ‘y) ∈ z ↔ ∀y ∈ x y ∈ z))
11		dfss3 1498	. . . . . . . . 9 ⊢ (x ⊆ z ↔ ∀y ∈ x y ∈ z)
12	10, 11	syl6bbr 416	. . . . . . . 8 ⊢ (∀y ∈ x (rank ‘y) = y → (∀y ∈ x (rank ‘y) ∈ z ↔ x ⊆ z))
13	12	birabsdv 1344	. . . . . . 7 ⊢ (∀y ∈ x (rank ‘y) = y → {z ∈ On∣∀y ∈ x (rank ‘y) ∈ z} = {z ∈ On∣x ⊆ z})
14	13	inteqd 1970	. . . . . 6 ⊢ (∀y ∈ x (rank ‘y) = y → ∩{z ∈ On∣∀y ∈ x (rank ‘y) ∈ z} = ∩{z ∈ On∣x ⊆ z})
15		visset 1350	. . . . . . 7 ⊢ x ∈ V
16	15	rankval3 3525	. . . . . 6 ⊢ (rank ‘x) = ∩{z ∈ On∣∀y ∈ x (rank ‘y) ∈ z}
17	14, 16	syl5eq 1136	. . . . 5 ⊢ (∀y ∈ x (rank ‘y) = y → (rank ‘x) = ∩{z ∈ On∣x ⊆ z})
18		intmin 1982	. . . . . 6 ⊢ (x ∈ On → x = ∩{z ∈ On∣x ⊆ z})
19	18	cleqcomd 1106	. . . . 5 ⊢ (x ∈ On → ∩{z ∈ On∣x ⊆ z} = x)
20	17, 19	sylan9eqr 1145	. . . 4 ⊢ ((x ∈ On ∧ ∀y ∈ x (rank ‘y) = y) → (rank ‘x) = x)
21	20	exp 291	. . 3 ⊢ (x ∈ On → (∀y ∈ x (rank ‘y) = y → (rank ‘x) = x))
22	3, 6, 21	tfis3 2248	. 2 ⊢ (A ∈ On → (rank ‘A) = A)
23		rankon 3515	. . 3 ⊢ (rank ‘A) ∈ On
24		eleq1 1149	. . 3 ⊢ ((rank ‘A) = A → ((rank ‘A) ∈ On ↔ A ∈ On))
25	23, 24	mpbii 168	. 2 ⊢ ((rank ‘A) = A → A ∈ On)
26	22, 25	impbi 139	1 ⊢ (A ∈ On ↔ (rank ‘A) = A)

Syntax hints:   ↔ wb 127   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204   ⊆ wss 1487  ∩cint 1965  Oncon0 2199   ‘cfv 2422  rankcrnk 3486

This theorem is referenced by:  rankr1id 3539

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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