Theorem bndrank 4692

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:   x,y,A

Proof of Theorem bndrank

Step	Hyp	Ref	Expression
1		visset 1816	. . . . . 6 |- y e. V
2	1	rankid 4682	. . . . 5 |- y e. (R1` suc (rank` y))
3		eloni 2964	. . . . . . . 8 |- (x e. On -> Ord x)
4		rankon 4681	. . . . . . . . . 10 |- (rank` y) e. On
5	4	onord 3101	. . . . . . . . 9 |- Ord (rank` y)
6		ordsucsssuc 3080	. . . . . . . . 9 |- ((Ord (rank` y) /\ Ord x) -> ((rank` y) (_ x <-> suc (rank` y) (_ suc x))
7	5, 6	mpan 697	. . . . . . . 8 |- (Ord x -> ((rank` y) (_ x <-> suc (rank` y) (_ suc x))
8	3, 7	syl 10	. . . . . . 7 |- (x e. On -> ((rank` y) (_ x <-> suc (rank` y) (_ suc x))
9		suceloni 3068	. . . . . . . 8 |- (x e. On -> suc x e. On)
10	4	onsuc 3111	. . . . . . . . 9 |- suc (rank` y) e. On
11		r1ord3 4667	. . . . . . . . 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 697	. . . . . . . 8 |- (suc x e. On -> (suc (rank` y) (_ suc x -> (R1` suc (rank` y)) (_ (R1` suc x)))
13	9, 12	syl 10	. . . . . . 7 |- (x e. On -> (suc (rank` y) (_ suc x -> (R1` suc (rank` y)) (_ (R1` suc x)))
14	8, 13	sylbid 203	. . . . . 6 |- (x e. On -> ((rank` y) (_ x -> (R1` suc (rank` y)) (_ (R1` suc x)))
15		ssel 2066	. . . . . 6 |- ((R1` suc (rank` y)) (_ (R1` suc x) -> (y e. (R1` suc (rank` y)) -> y e. (R1` suc x)))
16	14, 15	syl6 22	. . . . 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 1713	. . 3 |- (x e. On -> (A.y e. A (rank` y) (_ x -> A.y e. A y e. (R1` suc x)))
19		dfss3 2062	. . . 4 |- (A (_ (R1` suc x) <-> A.y e. A y e. (R1` suc x))
20		fvex 3738	. . . . 5 |- (R1` suc x) e. V
21	20	ssex 2724	. . . 4 |- (A (_ (R1` suc x) -> A e. V)
22	19, 21	sylbir 201	. . 3 |- (A.y e. A y e. (R1` suc x) -> A e. V)
23	18, 22	syl6 22	. 2 |- (x e. On -> (A.y e. A (rank` y) (_ x -> A e. V))
24	23	r19.23aiv 1746	1 |- (E.x e. On A.y e. A (rank` y) (_ x -> A e. V)

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 960  A.wral 1648  E.wrex 1649  Vcvv 1814   (_ wss 2050  Ord word 2953  Oncon0 2954  suc csuc 2956  ` cfv 3188  R1cr1 4651  rankcrnk 4652

This theorem is referenced by:  unbndrank 4693  scottex 4726

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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