Theorem bndrank 4662

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 1809	. . . . . 6 |- y e. V
2	1	rankid 4652	. . . . 5 |- y e. (R1` suc (rank` y))
3		eloni 2953	. . . . . . . 8 |- (x e. On -> Ord x)
4		rankon 4651	. . . . . . . . . 10 |- (rank` y) e. On
5	4	onord 3090	. . . . . . . . 9 |- Ord (rank` y)
6		ordsucsssuc 3069	. . . . . . . . 9 |- ((Ord (rank` y) /\ Ord x) -> ((rank` y) (_ x <-> suc (rank` y) (_ suc x))
7	5, 6	mpan 694	. . . . . . . 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 3057	. . . . . . . 8 |- (x e. On -> suc x e. On)
10	4	onsuc 3100	. . . . . . . . 9 |- suc (rank` y) e. On
11		r1ord3 4637	. . . . . . . . 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 694	. . . . . . . 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 2059	. . . . . 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 1707	. . 3 |- (x e. On -> (A.y e. A (rank` y) (_ x -> A.y e. A y e. (R1` suc x)))
19		dfss3 2055	. . . 4 |- (A (_ (R1` suc x) <-> A.y e. A y e. (R1` suc x))
20		fvex 3723	. . . . 5 |- (R1` suc x) e. V
21	20	ssex 2714	. . . 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 1740	1 |- (E.x e. On A.y e. A (rank` y) (_ x -> A e. V)

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 956  A.wral 1642  E.wrex 1643  Vcvv 1807   (_ wss 2043  Ord word 2942  Oncon0 2943  suc csuc 2945  ` cfv 3177  R1cr1 4621  rankcrnk 4622

This theorem is referenced by:  unbndrank 4663  scottex 4696

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623  df-rank 4624

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