Theorem rankxplim 4722

Description: The rank of a cross product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 4725 for the successor case.

Hypotheses

Ref	Expression
rankxplim.1	|- A e. V
rankxplim.2	|- B e. V

Assertion

Ref	Expression
rankxplim	|- ((Lim (rank` (A u. B)) /\ (A X. B) =/= (/)) -> (rank` (A X. B)) = (rank` (A u. B)))

Proof of Theorem rankxplim

Step	Hyp	Ref	Expression
1		visset 1816	. . . . . . . . . . . 12 |- x e. V
2		visset 1816	. . . . . . . . . . . 12 |- y e. V
3		rankxplim.1	. . . . . . . . . . . 12 |- A e. V
4		rankxplim.2	. . . . . . . . . . . 12 |- B e. V
5	1, 2, 3, 4	rankelun 4717	. . . . . . . . . . 11 |- (((rank` x) e. (rank` A) /\ (rank` y) e. (rank` B)) -> (rank` (x u. y)) e. (rank` (A u. B)))
6	3	rankel 4690	. . . . . . . . . . 11 |- (x e. A -> (rank` x) e. (rank` A))
7	4	rankel 4690	. . . . . . . . . . 11 |- (y e. B -> (rank` y) e. (rank` B))
8	5, 6, 7	syl2an 456	. . . . . . . . . 10 |- ((x e. A /\ y e. B) -> (rank` (x u. y)) e. (rank` (A u. B)))
9	8	adantl 390	. . . . . . . . 9 |- ((Lim (rank` (A u. B)) /\ (x e. A /\ y e. B)) -> (rank` (x u. y)) e. (rank` (A u. B)))
10		ranklim 4695	. . . . . . . . . . 11 |- (Lim (rank` (A u. B)) -> ((rank` (x u. y)) e. (rank` (A u. B)) <-> (rank` P~(x u. y)) e. (rank` (A u. B))))
11		ranklim 4695	. . . . . . . . . . 11 |- (Lim (rank` (A u. B)) -> ((rank` P~(x u. y)) e. (rank` (A u. B)) <-> (rank` P~P~(x u. y)) e. (rank` (A u. B))))
12	10, 11	bitrd 530	. . . . . . . . . 10 |- (Lim (rank` (A u. B)) -> ((rank` (x u. y)) e. (rank` (A u. B)) <-> (rank` P~P~(x u. y)) e. (rank` (A u. B))))
13	12	adantr 391	. . . . . . . . 9 |- ((Lim (rank` (A u. B)) /\ (x e. A /\ y e. B)) -> ((rank` (x u. y)) e. (rank` (A u. B)) <-> (rank` P~P~(x u. y)) e. (rank` (A u. B))))
14	9, 13	mpbid 195	. . . . . . . 8 |- ((Lim (rank` (A u. B)) /\ (x e. A /\ y e. B)) -> (rank` P~P~(x u. y)) e. (rank` (A u. B)))
15		pwuni 2763	. . . . . . . . . . . 12 |- <.x, y>. (_ P~U.<.x, y>.
16		uniop 2814	. . . . . . . . . . . . 13 |- U.<.x, y>. = {x, y}
17		pweq 2407	. . . . . . . . . . . . 13 |- (U.<.x, y>. = {x, y} -> P~U.<.x, y>. = P~{x, y})
18	16, 17	ax-mp 7	. . . . . . . . . . . 12 |- P~U.<.x, y>. = P~{x, y}
19	15, 18	sseqtr 2096	. . . . . . . . . . 11 |- <.x, y>. (_ P~{x, y}
20		pwuni 2763	. . . . . . . . . . . . 13 |- {x, y} (_ P~U.{x, y}
21	1, 2	unipr 2519	. . . . . . . . . . . . . 14 |- U.{x, y} = (x u. y)
22		pweq 2407	. . . . . . . . . . . . . 14 |- (U.{x, y} = (x u. y) -> P~U.{x, y} = P~(x u. y))
23	21, 22	ax-mp 7	. . . . . . . . . . . . 13 |- P~U.{x, y} = P~(x u. y)
24	20, 23	sseqtr 2096	. . . . . . . . . . . 12 |- {x, y} (_ P~(x u. y)
25		sspwb 2761	. . . . . . . . . . . 12 |- ({x, y} (_ P~(x u. y) <-> P~{x, y} (_ P~P~(x u. y))
26	24, 25	mpbi 189	. . . . . . . . . . 11 |- P~{x, y} (_ P~P~(x u. y)
27	19, 26	sstri 2076	. . . . . . . . . 10 |- <.x, y>. (_ P~P~(x u. y)
28	1, 2	unex 2878	. . . . . . . . . . . . 13 |- (x u. y) e. V
29	28	pwex 2751	. . . . . . . . . . . 12 |- P~(x u. y) e. V
30	29	pwex 2751	. . . . . . . . . . 11 |- P~P~(x u. y) e. V
31	30	rankss 4698	. . . . . . . . . 10 |- (<.x, y>. (_ P~P~(x u. y) -> (rank` <.x, y>.) (_ (rank` P~P~(x u. y)))
32	27, 31	ax-mp 7	. . . . . . . . 9 |- (rank` <.x, y>.) (_ (rank` P~P~(x u. y))
33		rankon 4681	. . . . . . . . . 10 |- (rank` <.x, y>.) e. On
34		rankon 4681	. . . . . . . . . 10 |- (rank` (A u. B)) e. On
35		ontr2 3010	. . . . . . . . . 10 |- (((rank` <.x, y>.) e. On /\ (rank` (A u. B)) e. On) -> (((rank` <.x, y>.) (_ (rank` P~P~(x u. y)) /\ (rank` P~P~(x u. y)) e. (rank` (A u. B))) -> (rank` <.x, y>.) e. (rank` (A u. B))))
36	33, 34, 35	mp2an 699	. . . . . . . . 9 |- (((rank` <.x, y>.) (_ (rank` P~P~(x u. y)) /\ (rank` P~P~(x u. y)) e. (rank` (A u. B))) -> (rank` <.x, y>.) e. (rank` (A u. B)))
37	32, 36	mpan 697	. . . . . . . 8 |- ((rank` P~P~(x u. y)) e. (rank` (A u. B)) -> (rank` <.x, y>.) e. (rank` (A u. B)))
38	14, 37	syl 10	. . . . . . 7 |- ((Lim (rank` (A u. B)) /\ (x e. A /\ y e. B)) -> (rank` <.x, y>.) e. (rank` (A u. B)))
39	33, 34	onsucss 3117	. . . . . . 7 |- ((rank` <.x, y>.) e. (rank` (A u. B)) <-> suc (rank` <.x, y>.) (_ (rank` (A u. B)))
40	38, 39	sylib 198	. . . . . 6 |- ((Lim (rank` (A u. B)) /\ (x e. A /\ y e. B)) -> suc (rank` <.x, y>.) (_ (rank` (A u. B)))
41	40	ex 373	. . . . 5 |- (Lim (rank` (A u. B)) -> ((x e. A /\ y e. B) -> suc (rank` <.x, y>.) (_ (rank` (A u. B))))
42	41	r19.21aivv 1723	. . . 4 |- (Lim (rank` (A u. B)) -> A.x e. A A.y e. B suc (rank` <.x, y>.) (_ (rank` (A u. B)))
43		fveq2 3730	. . . . . . . 8 |- (z = <.x, y>. -> (rank` z) = (rank` <.x, y>.))
44		suceq 3040	. . . . . . . 8 |- ((rank` z) = (rank` <.x, y>.) -> suc (rank` z) = suc (rank` <.x, y>.))
45	43, 44	syl 10	. . . . . . 7 |- (z = <.x, y>. -> suc (rank` z) = suc (rank` <.x, y>.))
46	45	sseq1d 2091	. . . . . 6 |- (z = <.x, y>. -> (suc (rank` z) (_ (rank` (A u. B)) <-> suc (rank` <.x, y>.) (_ (rank` (A u. B))))
47	46	ralxp 3224	. . . . 5 |- (A.z e. (A X. B)suc (rank` z) (_ (rank` (A u. B)) <-> A.x e. A A.y e. B suc (rank` <.x, y>.) (_ (rank` (A u. B)))
48	3, 4	xpex 3266	. . . . . 6 |- (A X. B) e. V
49	48	rankbnd 4713	. . . . 5 |- (A.z e. (A X. B)suc (rank` z) (_ (rank` (A u. B)) <-> (rank` (A X. B)) (_ (rank` (A u. B)))
50	47, 49	bitr3 175	. . . 4 |- (A.x e. A A.y e. B suc (rank` <.x, y>.) (_ (rank` (A u. B)) <-> (rank` (A X. B)) (_ (rank` (A u. B)))
51	42, 50	sylib 198	. . 3 |- (Lim (rank` (A u. B)) -> (rank` (A X. B)) (_ (rank` (A u. B)))
52	51	adantr 391	. 2 |- ((Lim (rank` (A u. B)) /\ (A X. B) =/= (/)) -> (rank` (A X. B)) (_ (rank` (A u. B)))
53	3, 4	rankxpl 4720	. . 3 |- ((A X. B) =/= (/) -> (rank` (A u. B)) (_ (rank` (A X. B)))
54	53	adantl 390	. 2 |- ((Lim (rank` (A u. B)) /\ (A X. B) =/= (/)) -> (rank` (A u. B)) (_ (rank` (A X. B)))
55	52, 54	eqssd 2082	1 |- ((Lim (rank` (A u. B)) /\ (A X. B) =/= (/)) -> (rank` (A X. B)) = (rank` (A u. B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  Vcvv 1814   u. cun 2048   (_ wss 2050  (/)c0 2283  P~cpw 2405  {cpr 2414  <.cop 2415  U.cuni 2507  Oncon0 2954  Lim wlim 2955  suc csuc 2956   X. cxp 3174  ` cfv 3188  rankcrnk 4652

This theorem is referenced by:  rankxplim3 4724

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

Copyright terms: Public domain