Theorem bndrank 3526

Description: Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80.

Assertion

Ref	Expression
bndrank	|- (E.x e. On A.y e. A (rank` y) (_ x -> A e. V)

Distinct variable group(s):   x,y,A

Proof of Theorem bndrank

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . 6 |- y e. V
2	1	rankid 3516	. . . . 5 |- y e. (R1` suc (rank` y))
3		eloni 2209	. . . . . . . 8 |- (x e. On -> Ord x)
4		rankon 3515	. . . . . . . . . 10 |- (rank` y) e. On
5	4	onord 2343	. . . . . . . . 9 |- Ord (rank` y)
6		ordsucsssuc 2325	. . . . . . . . 9 |- ((Ord (rank` y) /\ Ord x) -> ((rank` y) (_ x <-> suc (rank` y) (_ suc x))
7	5, 6	mpan 518	. . . . . . . 8 |- (Ord x -> ((rank` y) (_ x <-> suc (rank` y) (_ suc x))
8	3, 7	syl 12	. . . . . . 7 |- (x e. On -> ((rank` y) (_ x <-> suc (rank` y) (_ suc x))
9		suceloni 2314	. . . . . . . 8 |- (x e. On -> suc x e. On)
10	4	onsuc 2353	. . . . . . . . 9 |- suc (rank` y) e. On
11		r1ord3 3501	. . . . . . . . 9 |- ((suc (rank` y) e. On /\ suc x e. On) -> (suc (rank` y) (_ suc x -> (R1` suc (rank` y)) (_ (R1` suc x)))
12	10, 11	mpan 518	. . . . . . . 8 |- (suc x e. On -> (suc (rank` y) (_ suc x -> (R1` suc (rank` y)) (_ (R1` suc x)))
13	9, 12	syl 12	. . . . . . 7 |- (x e. On -> (suc (rank` y) (_ suc x -> (R1` suc (rank` y)) (_ (R1` suc x)))
14	8, 13	sylbid 178	. . . . . 6 |- (x e. On -> ((rank` y) (_ x -> (R1` suc (rank` y)) (_ (R1` suc x)))
15		ssel 1502	. . . . . 6 |- ((R1` suc (rank` y)) (_ (R1` suc x) -> (y e. (R1` suc (rank` y)) -> y e. (R1` suc x)))
16	14, 15	syl6 23	. . . . 5 |- (x e. On -> ((rank` y) (_ x -> (y e. (R1` suc (rank` y)) -> y e. (R1` suc x))))
17	2, 16	mpii 45	. . . 4 |- (x e. On -> ((rank` y) (_ x -> y e. (R1` suc x)))
18	17	r19.20sdv 1257	. . 3 |- (x e. On -> (A.y e. A (rank` y) (_ x -> A.y e. A y e. (R1` suc x)))
19		dfss3 1498	. . . 4 |- (A (_ (R1` suc x) <-> A.y e. A y e. (R1` suc x))
20		fvex 2838	. . . . 5 |- (R1` suc x) e. V
21	20	ssex 1700	. . . 4 |- (A (_ (R1` suc x) -> A e. V)
22	19, 21	sylbir 176	. . 3 |- (A.y e. A y e. (R1` suc x) -> A e. V)
23	18, 22	syl6 23	. 2 |- (x e. On -> (A.y e. A (rank` y) (_ x -> A e. V))
24	23	r19.23aiv 1284	1 |- (E.x e. On A.y e. A (rank` y) (_ x -> A e. V)

Syntax hints:   -> wi 2   <-> wb 127   e. wcel 1092  A.wral 1201  E.wrex 1202  Vcvv 1348   (_ wss 1487  Ord word 2198  Oncon0 2199  suc csuc 2201  ` cfv 2422  R1cr1 3485  rankcrnk 3486

This theorem is referenced by:  unbndrank 3527  scottex 3541

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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