Theorem rankr1id 3539

Description: The rank of the hierarchy of an ordinal number is itself.

Assertion

Ref	Expression
rankr1id	⊢ (A ∈ On ↔ (rank ‘(R1 ‘A)) = A)

Proof of Theorem rankr1id

Step	Hyp	Ref	Expression
1		fveq2 2832	. . . . 5 ⊢ (x = A → (R1 ‘x) = (R1 ‘A))
2	1	fveq2d 2836	. . . 4 ⊢ (x = A → (rank ‘(R1 ‘x)) = (rank ‘(R1 ‘A)))
3		id 9	. . . 4 ⊢ (x = A → x = A)
4	2, 3	cleq12d 1115	. . 3 ⊢ (x = A → ((rank ‘(R1 ‘x)) = x ↔ (rank ‘(R1 ‘A)) = A))
5		r1ord3 3501	. . . . . . . . . 10 ⊢ ((x ∈ On ∧ y ∈ On) → (x ⊆ y → (R1 ‘x) ⊆ (R1 ‘y)))
6	5	exp 291	. . . . . . . . 9 ⊢ (x ∈ On → (y ∈ On → (x ⊆ y → (R1 ‘x) ⊆ (R1 ‘y))))
7	6	r19.21aiv 1259	. . . . . . . 8 ⊢ (x ∈ On → ∀y ∈ On (x ⊆ y → (R1 ‘x) ⊆ (R1 ‘y)))
8		ss2rab 1553	. . . . . . . 8 ⊢ ({y ∈ On∣x ⊆ y} ⊆ {y ∈ On∣(R1 ‘x) ⊆ (R1 ‘y)} ↔ ∀y ∈ On (x ⊆ y → (R1 ‘x) ⊆ (R1 ‘y)))
9	7, 8	sylibr 175	. . . . . . 7 ⊢ (x ∈ On → {y ∈ On∣x ⊆ y} ⊆ {y ∈ On∣(R1 ‘x) ⊆ (R1 ‘y)})
10		intss 1983	. . . . . . 7 ⊢ ({y ∈ On∣x ⊆ y} ⊆ {y ∈ On∣(R1 ‘x) ⊆ (R1 ‘y)} → ∩{y ∈ On∣(R1 ‘x) ⊆ (R1 ‘y)} ⊆ ∩{y ∈ On∣x ⊆ y})
11	9, 10	syl 12	. . . . . 6 ⊢ (x ∈ On → ∩{y ∈ On∣(R1 ‘x) ⊆ (R1 ‘y)} ⊆ ∩{y ∈ On∣x ⊆ y})
12		intmin 1982	. . . . . 6 ⊢ (x ∈ On → x = ∩{y ∈ On∣x ⊆ y})
13	11, 12	sseqtr4d 1537	. . . . 5 ⊢ (x ∈ On → ∩{y ∈ On∣(R1 ‘x) ⊆ (R1 ‘y)} ⊆ x)
14		fvex 2838	. . . . . 6 ⊢ (R1 ‘x) ∈ V
15		rankval2 3514	. . . . . 6 ⊢ ((R1 ‘x) ∈ V → (rank ‘(R1 ‘x)) = ∩{y ∈ On∣(R1 ‘x) ⊆ (R1 ‘y)})
16	14, 15	ax-mp 6	. . . . 5 ⊢ (rank ‘(R1 ‘x)) = ∩{y ∈ On∣(R1 ‘x) ⊆ (R1 ‘y)}
17	13, 16	syl5ss 1544	. . . 4 ⊢ (x ∈ On → (rank ‘(R1 ‘x)) ⊆ x)
18		rankonid 3538	. . . . 5 ⊢ (x ∈ On ↔ (rank ‘x) = x)
19		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
20		r1rankid 3537	. . . . . . . . 9 ⊢ (x ∈ V → x ⊆ (R1 ‘(rank ‘x)))
21	19, 20	ax-mp 6	. . . . . . . 8 ⊢ x ⊆ (R1 ‘(rank ‘x))
22		fveq2 2832	. . . . . . . . 9 ⊢ ((rank ‘x) = x → (R1 ‘(rank ‘x)) = (R1 ‘x))
23	22	sseq2d 1528	. . . . . . . 8 ⊢ ((rank ‘x) = x → (x ⊆ (R1 ‘(rank ‘x)) ↔ x ⊆ (R1 ‘x)))
24	21, 23	mpbii 168	. . . . . . 7 ⊢ ((rank ‘x) = x → x ⊆ (R1 ‘x))
25	14	rankss 3531	. . . . . . 7 ⊢ (x ⊆ (R1 ‘x) → (rank ‘x) ⊆ (rank ‘(R1 ‘x)))
26	24, 25	syl 12	. . . . . 6 ⊢ ((rank ‘x) = x → (rank ‘x) ⊆ (rank ‘(R1 ‘x)))
27		sseq1 1521	. . . . . 6 ⊢ ((rank ‘x) = x → ((rank ‘x) ⊆ (rank ‘(R1 ‘x)) ↔ x ⊆ (rank ‘(R1 ‘x))))
28	26, 27	mpbid 170	. . . . 5 ⊢ ((rank ‘x) = x → x ⊆ (rank ‘(R1 ‘x)))
29	18, 28	sylbi 174	. . . 4 ⊢ (x ∈ On → x ⊆ (rank ‘(R1 ‘x)))
30	17, 29	eqssd 1518	. . 3 ⊢ (x ∈ On → (rank ‘(R1 ‘x)) = x)
31	4, 30	vtoclga 1387	. 2 ⊢ (A ∈ On → (rank ‘(R1 ‘A)) = A)
32		rankon 3515	. . 3 ⊢ (rank ‘(R1 ‘A)) ∈ On
33		eleq1 1149	. . 3 ⊢ ((rank ‘(R1 ‘A)) = A → ((rank ‘(R1 ‘A)) ∈ On ↔ A ∈ On))
34	32, 33	mpbii 168	. 2 ⊢ ((rank ‘(R1 ‘A)) = A → A ∈ On)
35	31, 34	impbi 139	1 ⊢ (A ∈ On ↔ (rank ‘(R1 ‘A)) = A)

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∩cint 1965  Oncon0 2199   ‘cfv 2422  R₁cr1 3485  rankcrnk 3486

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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