Theorem ranksn 3532

Description: The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112.

Hypothesis

Ref	Expression
ranksn.1	⊢ A ∈ V

Assertion

Ref	Expression
ranksn	⊢ (rank ‘{A}) = suc (rank ‘A)

Proof of Theorem ranksn

Step	Hyp	Ref	Expression
1		df-ral 1205	. . . . . 6 ⊢ (∀y ∈ {A} (rank ‘y) ∈ x ↔ ∀y(y ∈ {A} → (rank ‘y) ∈ x))
2		elsn 1820	. . . . . . . 8 ⊢ (y ∈ {A} ↔ y = A)
3	2	imbi1i 161	. . . . . . 7 ⊢ ((y ∈ {A} → (rank ‘y) ∈ x) ↔ (y = A → (rank ‘y) ∈ x))
4	3	bial 695	. . . . . 6 ⊢ (∀y(y ∈ {A} → (rank ‘y) ∈ x) ↔ ∀y(y = A → (rank ‘y) ∈ x))
5		ranksn.1	. . . . . . 7 ⊢ A ∈ V
6		fveq2 2832	. . . . . . . 8 ⊢ (y = A → (rank ‘y) = (rank ‘A))
7	6	eleq1d 1155	. . . . . . 7 ⊢ (y = A → ((rank ‘y) ∈ x ↔ (rank ‘A) ∈ x))
8	5, 7	ceqsalv 1364	. . . . . 6 ⊢ (∀y(y = A → (rank ‘y) ∈ x) ↔ (rank ‘A) ∈ x)
9	1, 4, 8	3bitr 155	. . . . 5 ⊢ (∀y ∈ {A} (rank ‘y) ∈ x ↔ (rank ‘A) ∈ x)
10	9	a1i 7	. . . 4 ⊢ (x ∈ On → (∀y ∈ {A} (rank ‘y) ∈ x ↔ (rank ‘A) ∈ x))
11	10	birabi 1342	. . 3 ⊢ {x ∈ On∣∀y ∈ {A} (rank ‘y) ∈ x} = {x ∈ On∣(rank ‘A) ∈ x}
12	11	inteqi 1969	. 2 ⊢ ∩{x ∈ On∣∀y ∈ {A} (rank ‘y) ∈ x} = ∩{x ∈ On∣(rank ‘A) ∈ x}
13		snex 1859	. . 3 ⊢ {A} ∈ V
14	13	rankval3 3525	. 2 ⊢ (rank ‘{A}) = ∩{x ∈ On∣∀y ∈ {A} (rank ‘y) ∈ x}
15		rankon 3515	. . 3 ⊢ (rank ‘A) ∈ On
16		onsucmin 2323	. . 3 ⊢ ((rank ‘A) ∈ On → suc (rank ‘A) = ∩{x ∈ On∣(rank ‘A) ∈ x})
17	15, 16	ax-mp 6	. 2 ⊢ suc (rank ‘A) = ∩{x ∈ On∣(rank ‘A) ∈ x}
18	12, 14, 17	3eqtr4 1126	1 ⊢ (rank ‘{A}) = suc (rank ‘A)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204  Vcvv 1348  {csn 1808  ∩cint 1965  Oncon0 2199  suc csuc 2201   ‘cfv 2422  rankcrnk 3486

This theorem is referenced by:  rankpr 3536

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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