Theorem rankbnd2 4714

Description: The rank of a set is bounded by the successor of a bound for its members.

Hypothesis

Ref	Expression
rankr1b.1	|- A e. V

Assertion

Ref	Expression
rankbnd2	|- (B e. On -> (A.x e. A (rank` x) (_ B <-> (rank` A) (_ suc B))

Distinct variable groups:   x,A   x,B

Proof of Theorem rankbnd2

Step	Hyp	Ref	Expression
1		eloni 2964	. . 3 |- (B e. On -> Ord B)
2		rankon 4681	. . . . 5 |- (rank` A) e. On
3	2	onss 3105	. . . 4 |- (rank` A) (_ On
4		ordunisssuc 3089	. . . 4 |- (((rank` A) (_ On /\ Ord B) -> (U.(rank` A) (_ B <-> (rank` A) (_ suc B))
5	3, 4	mpan 697	. . 3 |- (Ord B -> (U.(rank` A) (_ B <-> (rank` A) (_ suc B))
6	1, 5	syl 10	. 2 |- (B e. On -> (U.(rank` A) (_ B <-> (rank` A) (_ suc B))
7		rankuni 4708	. . . . 5 |- (rank` U.A) = U.(rank` A)
8		rankr1b.1	. . . . . 6 |- A e. V
9	8	rankuni2 4700	. . . . 5 |- (rank` U.A) = U_x e. A (rank` x)
10	7, 9	eqtr3 1500	. . . 4 |- U.(rank` A) = U_x e. A (rank` x)
11	10	sseq1i 2088	. . 3 |- (U.(rank` A) (_ B <-> U_x e. A (rank` x) (_ B)
12		iunss 2595	. . 3 |- (U_x e. A (rank` x) (_ B <-> A.x e. A (rank` x) (_ B)
13	11, 12	bitr2 174	. 2 |- (A.x e. A (rank` x) (_ B <-> U.(rank` A) (_ B)
14	6, 13	syl5bb 534	1 |- (B e. On -> (A.x e. A (rank` x) (_ B <-> (rank` A) (_ suc B))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 960  A.wral 1648  Vcvv 1814   (_ wss 2050  U.cuni 2507  U_ciun 2570  Ord word 2953  Oncon0 2954  suc csuc 2956  ` cfv 3188  rankcrnk 4652

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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